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TR11-028 | 24th February 2011
Richard Beigel, Bin Fu

#### A Dense Hierarchy of Sublinear Time Approximation Schemes for Bin Packing

The bin packing problem is to find the minimum
number of bins of size one to pack a list of items with sizes
$a_1,\ldots , a_n$ in $(0,1]$. Using uniform sampling, which selects
a random element from the input list each time, we develop a
N)$deterministic lower bound fails ... more >>> TR94-027 | 12th December 1994 Stasys Jukna #### A Note on Read-k Times Branching Programs A syntactic read-k times branching program has the restriction that no variable occurs more than k times on any path (whether or not consistent). We exhibit an explicit Boolean function f which cannot be computed by nondeterministic syntactic read-k times branching programs of size less than exp(\sqrt{n}}k^{-2k}), ... more >>> TR95-009 | 2nd February 1995 Matthias Krause #### A note on realizing iterated multiplication by small depth threshold circuits It is shown that decomposition via Chinise Remainder does not yield polynomial size depth 3 threshold circuits for iterated multiplication of n-bit numbers. This result is achieved by proving that, in contrast to multiplication of two n-bit numbers, powering, division, and other related problems, the ... more >>> TR07-105 | 21st September 2007 Jelani Nelson #### A Note on Set Cover Inapproximability Independent of Universe Size Revisions: 1 In the set cover problem we are given a collection of$m$sets whose union covers$[n] = \{1,\ldots,n\}$and must find a minimum-sized subcollection whose union still covers$[n]$. We investigate the approximability of set cover by an approximation ratio that depends only on$m$and observe that, for ... more >>> TR01-029 | 27th March 2001 Denis Xavier Charles #### A Note on Subgroup Membership Problem for PSL(2,p). Comments: 1 We show that there are infinitely many primes$p$, such that the subgroup membership problem for PSL(2,p) belongs to$\NP \cap \coNP$. more >>> TR12-095 | 23rd July 2012 Avraham Ben-Aroya, Igor Shinkar #### A Note on Subspace Evasive Sets A subspace-evasive set over a field${\mathbb F}$is a subset of${\mathbb F}^n$that has small intersection with any low-dimensional affine subspace of${\mathbb F}^n$. Interest in subspace evasive sets began in the work of Pudlák and Rödl (Quaderni di Matematica 2004). More recently, Guruswami (CCC 2011) showed that ... more >>> TR05-130 | 31st October 2005 Ahuva Mu'alem #### A Note on Testing Truthfulness This work initiates the study of algorithms for the testing of monotonicity of mechanisms. Such testing algorithms are useful for searching dominant strategy mechanisms. An$\e$-tester for monotonicity is given a query access to a mechanism, accepts if monotonicity is satisfied, and rejects with high probability if more than$\e$-fraction more >>> TR04-056 | 1st July 2004 N. V. Vinodchandran #### A note on the circuit complexity of PP In this short note we show that for any integer k, there are languages in the complexity class PP that do not have Boolean circuits of size$n^k$. more >>> TR13-069 | 1st May 2013 Kousha Etessami, Alistair Stewart, Mihalis Yannakakis #### A note on the complexity of comparing succinctly represented integers, with an application to maximum probability parsing The following two decision problems capture the complexity of comparing integers or rationals that are succinctly represented in product-of-exponentials notation, or equivalently, via arithmetic circuits using only multiplication and division gates, and integer inputs: Input instance: four lists of positive integers:$a_1, \ldots , a_n; \ b_1, \ldots ,b_n; \ ... more >>>

TR01-092 | 2nd October 2001
Till Tantau

#### A Note on the Complexity of the Reachability Problem for Tournaments

Deciding whether a vertex in a graph is reachable from another
vertex has been studied intensively in complexity theory and is
well understood. For common types of graphs like directed graphs,
undirected graphs, dags or trees it takes a (possibly
nondeterministic) logspace machine to decide the reachability
problem, and ... more >>>

TR06-076 | 4th May 2006
Noam Nisan

#### A Note on the computational hardness of evolutionary stable strategies

We present a very simple reduction that when given a graph G and an integer k produces a game that has an evolutionary stable strategy if and only if the maximum clique size of G is not exactly k. Formally this shows that existence of evolutionary stable strategies is hard ... more >>>

TR08-012 | 20th November 2007
Arnab Bhattacharyya

#### A Note on the Distance to Monotonicity of Boolean Functions

Given a boolean function, let epsilon_M(f) denote the smallest distance between f and a monotone function on {0,1}^n. Let delta_M(f) denote the fraction of hypercube edges where f violates monotonicity. We give an alternative proof of the tight bound: delta_M(f) >= 2/n eps_M(f) for any boolean function f. This was ... more >>>

TR00-088 | 28th November 2000
Meena Mahajan, V. Vinay

#### A note on the hardness of the characteristic polynomial

In this note, we consider the problem of computing the
coefficients of the characteristic polynomial of a given
matrix, and the related problem of verifying the
coefficents.

Santha and Tan [CC98] show that verifying the determinant
(the constant term in the characteristic polynomial) is
complete for the class C=L, and ... more >>>

TR12-092 | 6th July 2012
Pavol Duris

#### A Note On the Hierarchy of One-way Data-Independent Multi-Head Finite Automata.

In this paper we deal with one-way multi-head data-independent finite automata. A $k$-head finite automaton $A$ is data-independent, if the position of every head $i$ after step $t$ in the computation on an input $w$ is a function that depends only on the length of the input $w$, on $i$ ... more >>>

TR01-058 | 28th August 2001
Stasys Jukna

#### A Note on the Minimum Number of Negations Leading to Superpolynomial Savings

In 1957 Markov proved that every circuit in $n$ variables
can be simulated by a circuit with at most $\log(n+1)$ negations.
In 1974 Fischer has shown that this can be done with only
polynomial increase in size.

In this note we observe that some explicit monotone functions
more >>>

TR04-062 | 28th July 2004
Stasys Jukna

#### A note on the P versus NP intersected with co-NP question in communication complexity

We consider the P versus NP\cap coNP question for the classical two-party communication protocols: if both a boolean function and its negation have small nondeterministic communication complexity, what is then its deterministic and/or probabilistic communication complexity? In the fixed (worst) partition case this question was answered by Aho, Ullman and ... more >>>

TR02-004 | 2nd November 2001
Till Tantau

#### A Note on the Power of Extra Queries to Membership Comparable Sets

A language is called k-membership comparable if there exists a
polynomial-time algorithm that excludes for any k words one of
the 2^k possibilities for their characteristic string.
It is known that all membership comparable languages can be
reduced to some P-selective language with polynomially many
adaptive queries. We show however ... more >>>

TR12-121 | 25th September 2012
Pavel Hrubes

#### A note on the real $\tau$-conjecture and the distribution of roots

Revisions: 2

Koiran's real $\tau$-conjecture asserts that if a non-zero real polynomial can be written as $f=\sum_{i=1}^{p}\prod_{j=1}^{q}f_{ij},$
where each $f_{ij}$ contains at most $k$ monomials, then the number of distinct real roots of $f$ is polynomial in $pqk$. We show that the conjecture implies quite a strong property of the ... more >>>

TR98-042 | 27th July 1998
Pavel Pudlak

#### A Note On the Use of Determinant for Proving Lower Bounds on the Size of Linear Circuits

We consider computations of linear forms over {\bf R} by
circuits with linear gates where the absolute values
coefficients are bounded by a constant. Also we consider a
related concept of restricted rigidity of a matrix. We prove
some lower bounds on the size of such circuits and the
restricted ... more >>>

TR04-118 | 21st December 2004
Marek Karpinski, Yakov Nekrich

#### A Note on Traversing Skew Merkle Trees

We consider the problem of traversing skew (unbalanced) Merkle
trees and design an algorithm for traversing a skew Merkle tree
in time O(log n/log t) and space O(log n (t/log t)), for any choice
of parameter t\geq 2.
This algorithm can be of special interest in situations when
more >>>

TR96-023 | 21st March 1996
Eric Allender

#### A Note on Uniform Circuit Lower Bounds for the Counting Hierarchy

A very recent paper by Caussinus, McKenzie, Therien, and Vollmer
[CMTV] shows that ACC^0 is properly contained in ModPH, and TC^0
is properly contained in the counting hierarchy. Thus, [CMTV] shows
that there are problems in ModPH that require superpolynomial-size
uniform ACC^0 ... more >>>

TR07-016 | 13th February 2007

#### A Note on Yekhanin's Locally Decodable Codes

Revisions: 1

Locally Decodable codes(LDC) support decoding of any particular symbol of the input message by reading constant number of symbols of the codeword, even in presence of constant fraction of errors.

In a recent breakthrough, Yekhanin designed $3$-query LDCs that hugely improve over earlier constructions. Specifically, for a Mersenne prime $p ... more >>> TR09-027 | 2nd April 2009 Iftach Haitner #### A Parallel Repetition Theorem for Any Interactive Argument Revisions: 1 The question whether or not parallel repetition reduces the soundness error is a fundamental question in the theory of protocols. While parallel repetition reduces (at an exponential rate) the error in interactive proofs and (at a weak exponential rate) in special cases of interactive arguments (e.g., 3-message protocols - Bellare, ... more >>> TR10-019 | 19th February 2010 Andrew Drucker #### A PCP Characterization of AM We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class$\mathsf{AM}$. This gives a PCP characterization' of$\mathsf{AM}$analogous to the PCP Theorem for$\mathsf{NP}$. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, ... more >>> TR00-064 | 29th August 2000 Klaus Jansen, Marek Karpinski, Andrzej Lingas #### A Polynomial Time Approximation Scheme for MAX-BISECTION on Planar Graphs The Max-Bisection and Min-Bisection are the problems of finding partitions of the vertices of a given graph into two equal size subsets so as to maximize or minimize, respectively, the number of edges with exactly one endpoint in each subset. In this paper we design the first ... more >>> TR02-041 | 2nd July 2002 Wenceslas Fernandez de la Vega, Marek Karpinski, Claire Kenyon #### A Polynomial Time Approximation Scheme for Metric MIN-BISECTION We design a polynomial time approximation scheme (PTAS) for the problem of Metric MIN-BISECTION of dividing a given finite metric space into two halves so as to minimize the sum of distances across that partition. The method of solution depends on a new metric placement partitioning ... more >>> TR02-044 | 16th July 2002 Wenceslas Fernandez de la Vega, Marek Karpinski #### A Polynomial Time Approximation Scheme for Subdense MAX-CUT We prove that the subdense instances of MAX-CUT of average degree Omega(n/logn) posses a polynomial time approximation scheme (PTAS). We extend this result also to show that the instances of general 2-ary maximum constraint satisfaction problems (MAX-CSP) of the same average density have PTASs. Our results ... more >>> TR10-088 | 17th May 2010 Jiri Sima, Stanislav Zak #### A Polynomial Time Construction of a Hitting Set for Read-Once Branching Programs of Width 3 Revisions: 2 , Comments: 3 The relationship between deterministic and probabilistic computations is one of the central issues in complexity theory. This problem can be tackled by constructing polynomial time hitting set generators which, however, belongs to the hardest problems in computer science even for severely restricted computational models. In our work, we consider read-once ... more >>> TR04-038 | 27th April 2004 John Case, Sanjay Jain, Rüdiger Reischuk, Frank Stephan, Thomas Zeugmann #### A Polynomial Time Learner for a Subclass of Regular Patterns Presented is an algorithm (for learning a subclass of erasing regular pattern languages) which can be made to run with arbitrarily high probability of success on extended regular languages generated by patterns$\pi$of the form$x_0 \alpha_1 x_1 ... \alpha_m x_m$for unknown$m$but known$c$, more >>> TR00-079 | 12th September 2000 Mark Jerrum, Eric Vigoda #### A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries We present a fully-polynomial randomized approximation scheme for computing the permanent of an arbitrary matrix with non-negative entries. more >>> TR09-078 | 16th September 2009 Falk Unger #### A Probabilistic Inequality with Applications to Threshold Direct-product Theorems We prove a simple concentration inequality, which is an extension of the Chernoff bound and Hoeffding's inequality for binary random variables. Instead of assuming independence of the variables we use a slightly weaker condition, namely bounds on the co-moments. This inequality allows us to simplify and strengthen several known direct-product ... more >>> TR98-051 | 20th July 1998 Petr Savicky #### A probabilistic nonequivalence test for syntactic (1,+k)-branching programs Branching programs are a model for representing Boolean functions. For general branching programs, the satisfiability and nonequivalence tests are NP-complete. For read-once branching programs, which can test each variable at most once in each computation, the satisfiability test is trivial and there is also a probabilistic polynomial time test of ... more >>> TR05-103 | 17th August 2005 Leonid Gurvits #### A proof of hyperbolic van der Waerden conjecture : the right generalization is the ultimate simplification Consider a homogeneous polynomial$p(z_1,...,z_n)$of degree$n$in$n$complex variables . Assume that this polynomial satisfies the property : \\$|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$on the domain$\{(z_1,...,z_n) : Re(z_i) \geq 0 , 1 \leq i \leq n \}$. \\ We prove that ... more >>> TR04-063 | 23rd July 2004 Yehuda Lindell, Benny Pinkas #### A Proof of Yao's Protocol for Secure Two-Party Computation Revisions: 1 In the mid 1980's, Yao presented a constant-round protocol for securely computing any two-party functionality in the presence of semi-honest adversaries (FOCS 1986). In this paper, we provide a complete description of Yao's protocol, along with a rigorous proof of security. Despite the importance of Yao's protocol to the field ... more >>> TR96-065 | 13th December 1996 Miklos Ajtai, Cynthia Dwork #### A Public-Key Cryptosystem with Worst-Case/Average-Case Equivalence Revisions: 1 , Comments: 1 We present a probabilistic public key cryptosystem which is secure unless the following worst-case lattice problem can be solved in polynomial time: "Find the shortest nonzero vector in an n dimensional lattice L where the shortest vector v is unique in the sense that any other vector whose length ... more >>> TR03-086 | 1st December 2003 Amos Beimel, Tal Malkin #### A Quantitative Approach to Reductions in Secure Computation Secure computation is one of the most fundamental cryptographic tasks. It is known that all functions can be computed securely in the information theoretic setting, given access to a black box for some complete function such as AND. However, without such a black box, not all functions can be securely ... more >>> TR08-017 | 16th December 2007 Thomas Watson, Dieter van Melkebeek #### A Quantum Time-Space Lower Bound for the Counting Hierarchy We obtain the first nontrivial time-space lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real$d$and every positive real$\epsilon$... more >>> TR09-074 | 10th September 2009 Suguru Tamaki, Yuichi Yoshida #### A Query Efficient Non-Adaptive Long Code Test with Perfect Completeness Long Code testing is a fundamental problem in the area of property testing and hardness of approximation. Long Code is a function of the form$f(x)=x_i$for some index$i$. In the Long Code testing, the problem is, given oracle access to a collection of Boolean functions, to decide whether ... more >>> TR05-156 | 13th December 2005 Jonathan A. Kelner, Daniel A. Spielman #### A Randomized Polynomial-Time Simplex Algorithm for Linear Programming (Preliminary Version) We present the first randomized polynomial-time simplex algorithm for linear programming. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input. We begin by reducing the input linear program to a special form in which we ... more >>> TR05-107 | 28th September 2005 Avi Wigderson, David Xiao #### A Randomness-Efficient Sampler for Matrix-valued Functions and Applications Revisions: 1 In this paper we give a randomness-efficient sampler for matrix-valued functions. Specifically, we show that a random walk on an expander approximates the recent Chernoff-like bound for matrix-valued functions of Ahlswede and Winter, in a manner which depends optimally on the spectral gap. The proof uses perturbation theory, and is ... more >>> TR12-124 | 29th September 2012 Massimo Lauria #### A rank lower bound for cutting planes proofs of Ramsey Theorem Ramsey Theorem is a cornerstone of combinatorics and logic. In its simplest formulation it says that there is a function$r$such that any simple graph with$r(k,s)$vertices contains either a clique of size$k$or an independent set of size$s$. We study the complexity of proving upper ... more >>> TR05-035 | 24th March 2005 Christian Glaßer, Stephen Travers, Klaus W. Wagner #### A Reducibility that Corresponds to Unbalanced Leaf-Language Classes We introduce the polynomial-time tree reducibility (ptt-reducibility). Our main result states that for languages$B$and$C$it holds that$B$ptt-reduces to$C$if and only if the unbalanced leaf-language class of$B$is robustly contained in the unbalanced leaf-language class of$C$. ... more >>> TR04-006 | 6th January 2004 Günter Hotz #### A remark on nondecidabilities of the initial value problem of ODEs We prove that it is not decidable on R-machines if for a fixed finite intervall [a,b) the solution of the initial value problems of systems of ordinary differetial equations have solutions over this interval. This result holds independly from assumptions about differentiability of the right sides of the ODEs. Futhermore ... more >>> TR12-005 | 13th January 2012 Periklis Papakonstantinou, Guang Yang #### A remark on one-wayness versus pseudorandomness Every pseudorandom generator is in particular a one-way function. If we only consider part of the output of the pseudorandom generator is this still one-way? Here is a general setting formalizing this question. Suppose$G:\{0,1\}^n\rightarrow \{0,1\}^{\ell(n)}$is a pseudorandom generator with stretch$\ell(n)> n$. Let$M_R\in\{0,1\}^{m(n)\times \ell(n)}$be a linear ... more >>> TR97-020 | 15th May 1997 Oded Goldreich #### A Sample of Samplers -- A Computational Perspective on Sampling (survey). We consider the problem of estimating the average of a huge set of values. That is, given oracle access to an arbitrary function$f:\{0,1\}^n\mapsto[0,1]$, we need to estimate$2^{-n} \sum_{x\in\{0,1\}^n} f(x)$upto an additive error of$\epsilon$. We are allowed to employ a randomized algorithm which may err ... more >>> TR12-071 | 29th May 2012 Kazuhisa Seto, Suguru Tamaki #### A Satisfiability Algorithm and Average-Case Hardness for Formulas over the Full Binary Basis We present a moderately exponential time algorithm for the satisfiability of Boolean formulas over the full binary basis. For formulas of size at most$cn$, our algorithm runs in time$2^{(1-\mu_c)n}$for some constant$\mu_c>0$. As a byproduct of the running time analysis of our algorithm, we get strong ... more >>> TR01-024 | 1st March 2001 Stephen Cook, Antonina Kolokolova #### A second-order system for polynomial-time reasoning based on Graedel's theorem We introduce a second-order system V_1-Horn of bounded arithmetic formalizing polynomial-time reasoning, based on Graedel's second-order Horn characterization of P. Our system has comprehension over P predicates (defined by Graedel's second-order Horn formulas), and only finitely many function symbols. Other systems of polynomial-time reasoning either ... more >>> TR00-027 | 11th April 2000 Pavol Duris, Juraj Hromkovic, Katsushi Inoue #### A Separation of Determinism, Las Vegas and Nondeterminism for Picture Recognition The investigation of the computational power of randomized computations is one of the central tasks of current complexity and algorithm theory. In this paper for the first time a "strong" separation between the power of determinism, Las Vegas randomization, and nondeterminism for a compu- ting model is proved. The computing ... more >>> TR10-060 | 5th April 2010 Dmitry Gavinsky, Alexander A. Sherstov #### A Separation of NP and coNP in Multiparty Communication Complexity We prove that NP$\ne$coNP and coNP$\nsubseteq$MA in the number-on-forehead model of multiparty communication complexity for up to$k=(1-\epsilon)\log n$players, where$\epsilon>0$is any constant. Specifically, we construct a function$F:(\zoon)^k\to\zoo$with co-nondeterministic complexity$O(\log n)$and Merlin-Arthur complexity$n^{\Omega(1)}$. The problem was open for$k\geq3$. more >>> TR98-045 | 17th July 1998 Detlef Sieling #### A Separation of Syntactic and Nonsyntactic (1,+k)-Branching Programs For (1,+k)-branching programs and read-k-times branching programs syntactic and nonsyntactic variants can be distinguished. The nonsyntactic variants correspond in a natural way to sequential computations with restrictions on reading the input while lower bound proofs are easier or only known for the syntactic variants. In this paper it is shown ... more >>> TR13-017 | 23rd January 2013 Pratik Worah #### A Short Excursion into Semi-Algebraic Hierarchies This brief survey gives a (roughly) self-contained overview of some complexity theoretic results about semi-algebraic proof systems and related hierarchies and the strong connections between them. The article is not intended to be a detailed survey on "Lift and Project" type optimization hierarchies (cf. Chlamtac and Tulsiani) or related proof ... more >>> TR13-012 | 16th January 2013 Hasan Abasi, Nader Bshouty #### A Simple Algorithm for Undirected Hamiltonicity We develop a new algebraic technique that gives a simple randomized algorithm for the simple$k$-path problem with the same complexity$O^*(1.657^k)$as in [A. Bj\"orklund. Determinant Sums for Undirected Hamiltonicity. FOCS 2010, pp. 173--182, (2010). A. Bj\"orklund, T. Husfeldt, P. Kaski, M. Koivisto. Narrow sieves for parameterized paths and ... more >>> TR08-093 | 1st October 2008 Cristopher Moore, Alexander Russell #### A simple constant-probability RP reduction from NP to Parity P The proof of Toda's celebrated theorem that the polynomial hierarchy is contained in$\P^\numP$relies on the fact that, under mild technical conditions on the complexity class$\mathcal{C}$, we have$\exists \,\mathcal{C} \subset \BP \cdot \oplus \,\mathcal{C}$. More concretely, there is a randomized reduction which transforms nonempty sets and the ... more >>> TR00-030 | 31st May 2000 #### A Simple Model for Neural Computation with Firing Rates and Firing Correlations A simple extension of standard neural network models is introduced that provides a model for neural computations that involve both firing rates and firing correlations. Such extension appears to be useful since it has been shown that firing correlations play a significant computational role in many biological neural systems. Standard ... more >>> TR08-081 | 11th September 2008 Alexander Razborov #### A simple proof of Bazzi's theorem In 1990, Linial and Nisan asked if any polylog-wise independent distribution fools any function in AC^0. In a recent remarkable development, Bazzi solved this problem for the case of DNF formulas. The aim of this note is to present a simplified version of his proof. more >>> TR07-114 | 28th September 2007 Jakob Nordström #### A Simplified Way of Proving Trade-off Results for Resolution We present a greatly simplified proof of the length-space trade-off result for resolution in Hertel and Pitassi (2007), and also prove a couple of other theorems in the same vein. We point out two important ingredients needed for our proofs to work, and discuss possible conclusions to be drawn regarding ... more >>> TR12-053 | 30th April 2012 Ankur Moitra #### A Singly-Exponential Time Algorithm for Computing Nonnegative Rank Here, we give an algorithm for deciding if the nonnegative rank of a matrix$M$of dimension$m \times n$is at most$r$which runs in time$(nm)^{O(r^2)}$. This is the first exact algorithm that runs in time singly-exponential in$r$. This algorithm (and earlier algorithms) are built on ... more >>> TR09-047 | 20th April 2009 Eli Ben-Sasson, Jakob Nordström #### A Space Hierarchy for k-DNF Resolution Comments: 1 The k-DNF resolution proof systems are a family of systems indexed by the integer k, where the kth member is restricted to operating with formulas in disjunctive normal form with all terms of bounded arity k (k-DNF formulas). This family was introduced in [Krajicek 2001] as an extension of the ... more >>> TR10-150 | 19th September 2010 Bjørn Kjos-Hanssen #### A strong law of computationally weak subsets We show that in the setting of fair-coin measure on the power set of the natural numbers, each sufficiently random set has an infinite subset that computes no random set. That is, there is an almost sure event$\mathcal A$such that if$X\in\mathcal A$then$X$has an infinite ... more >>> TR10-141 | 18th September 2010 (removed) Ran Raz #### A Strong Parallel Repetition Theorem for Projection Games on Expanders Reason: This paper has been remove on the author's behalf. Please note that TR10-142 is the corrected version. TR10-142 | 18th September 2010 Ran Raz, Ricky Rosen #### A Strong Parallel Repetition Theorem for Projection Games on Expanders The parallel repetition theorem states that for any Two Prover Game with value at most$1-\epsilon$(for$\epsilon<1/2$), the value of the game repeated$n$times in parallel is at most$(1-\epsilon^3)^{\Omega(n/s)}$, where$s$is the length of the answers of the two provers. For Projection Games, the bound on ... more >>> TR95-060 | 21st November 1995 Nader H. Bshouty #### A Subexponential Exact Learning Algorithm for DNF Using Equivalence Queries We present a$2^{\tilde O(\sqrt{n})}$time exact learning algorithm for polynomial size DNF using equivalence queries only. In particular, DNF is PAC-learnable in subexponential time under any distribution. This is the first subexponential time PAC-learning algorithm for DNF under any distribution. more >>> TR97-050 | 27th October 1997 Stanislav Zak #### A subexponential lower bound for branching programs restricted with regard to some semantic aspects Branching programs (b.p.s) or binary decision diagrams are a general graph-based model of sequential computation. The b.p.s of polynomial size are a nonuniform counterpart of LOG. Lower bounds for different kinds of restricted b.p.s are intensively investigated. The restrictions based on the number of tests of more >>> TR07-099 | 30th September 2007 Dieter van Melkebeek #### A Survey of Lower Bounds for Satisfiability and Related Problems Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a linear-time, logarithmic-space algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by ... more >>> TR12-051 | 25th April 2012 Dmitry Gavinsky, Shachar Lovett, Michael Saks, Srikanth Srinivasan #### A Tail Bound for Read-k Families of Functions We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables$Y_1,\ldots,Y_r$are arbitrary Boolean functions of independent random variables$X_1,\ldots,X_m$, modulo a restriction that every$X_i$influences at most$k$of the variables$Y_1,\ldots,Y_r$. more >>> TR10-145 | 21st September 2010 Ron Rothblum #### A Taxonomy of Enhanced Trapdoor Permutations Trapdoor permutations (TDPs) are among the most widely studied building blocks of cryptography. Despite the extensive body of work that has been dedicated to their study, in many setting and applications (enhanced) trapdoor permutations behave unexpectedly. In particular, a TDP may become easy to invert when the inverter is given ... more >>> TR09-075 | 17th September 2009 Oded Goldreich, Brendan Juba, Madhu Sudan #### A Theory of Goal-Oriented Communication Revisions: 1 , Comments: 1 We put forward a general theory of goal-oriented communication, where communication is not an end in itself, but rather a means to achieving some goals of the communicating parties. The goals can vary from setting to setting, and we provide a general framework for describing any such goal. In this ... more >>> TR98-034 | 23rd June 1998 Venkatesan Guruswami, Daniel Lewin and Madhu Sudan, Luca Trevisan #### A tight characterization of NP with 3 query PCPs It is known that there exists a PCP characterization of NP where the verifier makes 3 queries and has a {\em one-sided} error that is bounded away from 1; and also that 2 queries do not suffice for such a characterization. Thus PCPs with 3 queries ... more >>> TR11-086 | 2nd June 2011 Masaki Yamamoto #### A tighter lower bound on the circuit size of the hardest Boolean functions In [IPL2005], Frandsen and Miltersen improved bounds on the circuit size$L(n)$of the hardest Boolean function on$n$input bits: for some constant$c>0$: $\left(1+\frac{\log n}{n}-\frac{c}{n}\right) \frac{2^n}{n} \leq L(n) \leq \left(1+3\frac{\log n}{n}+\frac{c}{n}\right) \frac{2^n}{n}.$ In this note, we announce a modest ... more >>> TR04-073 | 9th July 2004 Henning Fernau #### A Top-Down Approach to Search-Trees: Improved Algorithmics for 3-Hitting Set In this paper, we show how to systematically improve on parameterized algorithms and their analysis, focusing on search-tree based algorithms for d-Hitting Set, especially for d=3. We concentrate on algorithms which are easy to implement, in contrast with the highly sophisticated algorithms which have been elsewhere designed to ... more >>> TR03-075 | 7th September 2003 Agostino Capponi #### A tutorial on the Deterministic two-party Communication Complexity Communication complexity is concerned with the question: how much information do the participants of a communication system need to exchange in order to perform certain tasks? The minimum number of bits that must be communicated is the deterministic communication complexity of$f$. This complexity measure was introduced by Yao \cite{1} ... more >>> TR01-016 | 22nd December 2000 Ran Canetti #### A unified framework for analyzing security of protocols Revisions: 3 Building on known definitions, we present a unified general framework for defining and analyzing security of cryptographic protocols. The framework allows specifying the security requirements of a large number of cryptographic tasks, such as signature, encryption, authentication, key exchange, commitment, oblivious transfer, zero-knowledge, secret sharing, general function evaluation, and ... more >>> TR10-161 | 25th October 2010 Arnab Bhattacharyya, Elena Grigorescu, Asaf Shapira #### A Unified Framework for Testing Linear-Invariant Properties The study of the interplay between the testability of properties of Boolean functions and the invariances acting on their domain which preserve the property was initiated by Kaufman and Sudan (STOC 2008). Invariance with respect to F_2-linear transformations is arguably the most common symmetry exhibited by natural properties of Boolean ... more >>> TR05-078 | 25th May 2005 Kooshiar Azimian, Javad Mohajeri, Mahmoud Salmasizadeh, Siamak Fayyaz #### A Verifiable Partial Key Escrow, Based on McCurley Encryption Scheme Revisions: 1 In this paper, firstly we propose two new concepts concerning the notion of key escrow encryption schemes: provable partiality and independency. Roughly speaking we say that a scheme has provable partiality if existing polynomial time algorithm for recovering the secret knowing escrowed information implies a polynomial time algorithm that can ... more >>> TR02-033 | 11th June 2002 Beate Bollig #### A very simple function that requires exponential size nondeterministic graph-driven read-once branching programs Branching programs are a well-established computation model for boolean functions, especially read-once branching programs (BP1s) have been studied intensively. A very simple function$f$in$n^2$variables is exhibited such that both the function$f$and its negation$\neg f$can be computed by$\Sigma^3_p$-circuits, the ... more >>> TR13-029 | 19th February 2013 Deeparnab Chakrabarty, C. Seshadhri #### A {\huge${o(n)}$} monotonicity tester for Boolean functions over the hypercube Given oracle access to a Boolean function$f:\{0,1\}^n \mapsto \{0,1\}$, we design a randomized tester that takes as input a parameter$\eps>0$, and outputs {\sf Yes} if the function is monotone, and outputs {\sf No} with probability$>2/3$, if the function is$\eps$-far from monotone. That is,$f$needs to ... more >>> TR13-030 | 20th February 2013 Elad Haramaty, Noga Ron-Zewi, Madhu Sudan #### Absolutely Sound Testing of Lifted Codes In this work we present a strong analysis of the testability of a broad, and to date the most interesting known, class of "affine-invariant'' codes. Affine-invariant codes are codes whose coordinates are associated with a vector space and are invariant under affine transformations of the coordinate space. Affine-invariant linear codes ... more >>> TR11-050 | 11th April 2011 Claus-Peter Schnorr #### Accelerated Slide- and LLL-Reduction Revisions: 7 Given an LLL-basis$B$of dimension$n= hk$we accelerate slide-reduction with blocksize$k$to run under a reasonable assjmption in \$\frac1{6} \, n^2 h \,\log_{1+\varepsilon} \, \alpha $\ local SVP-computations in dimension$k$, where$\alpha \ge \frac 43$measures the quality of the ... more >>> TR07-088 | 7th September 2007 Elad Hazan, C. Seshadhri #### Adaptive Algorithms for Online Decision Problems Revisions: 1 We study the notion of learning in an oblivious changing environment. Existing online learning algorithms which minimize regret are shown to converge to the average of all locally optimal solutions. We propose a new performance metric, strengthening the standard metric of regret, to capture convergence to locally optimal solutions, and ... more >>> TR06-042 | 16th March 2006 Amit Deshpande, Santosh Vempala #### Adaptive Sampling and Fast Low-Rank Matrix Approximation We prove that any real matrix$A$contains a subset of at most$4k/\eps + 2k \log(k+1)$rows whose span contains" a matrix of rank at most$k$with error only$(1+\eps)$times the error of the best rank-$k$approximation of$A$. This leads to an algorithm to find such ... more >>> TR96-051 | 1st October 1996 Richard Beigel, William Gasarch, Ming Li, Louxin Zhang #### Addition in$\log_2{n}$+ O(1)$ Steps on Average: A Simple Analysis

We demonstrate the use of Kolmogorov complexity in average case
analysis of algorithms through a classical example: adding two $n$-bit
numbers in $\ceiling{\log_2{n}}+2$ steps on average. We simplify the
analysis of Burks, Goldstine, and von Neumann in 1946 and
(in more complete forms) of Briley and of Schay.

more >>>

TR11-008 | 27th January 2011
Scott Aaronson, Andrew Drucker

#### Advice Coins for Classical and Quantum Computation

We study the power of classical and quantum algorithms equipped with nonuniform advice, in the form of a coin whose bias encodes useful information. This question takes on particular importance in the quantum case, due to a surprising result that we prove: a quantum finite automaton with just two states ... more >>>

TR11-120 | 6th September 2011
Thomas Watson

#### Advice Lower Bounds for the Dense Model Theorem

Revisions: 1

We prove a lower bound on the amount of nonuniform advice needed by black-box reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution $D$ that is $\delta$-dense in a distribution that is ... more >>>

TR10-044 | 12th March 2010
Eli Ben-Sasson, Swastik Kopparty

{\em Dispersers} and {\em extractors} for affine sources of dimension $d$ in $\mathbb F_p^n$ --- where $\mathbb F_p$ denotes the finite field of prime size $p$ --- are functions $f: \mathbb F_p^n \rightarrow \mathbb F_p$ that behave pseudorandomly when their domain is restricted to any particular affine space $S \subseteq ... more >>> TR11-061 | 18th April 2011 Neeraj Kayal #### Affine projections of polynomials Revisions: 1 An$m$-variate polynomial$f$is said to be an affine projection of some$n$-variate polynomial$g$if there exists an$n \times m$matrix$A$and an$n$-dimensional vector$b$such that$f(x) = g(A x + b)$. In other words, if$f$can be obtained by replacing each variable ... more >>> TR01-035 | 15th April 2001 Amir Shpilka #### Affine Projections of Symmetric Polynomials In this paper we introduce a new model for computing=20 polynomials - a depth 2 circuit with a symmetric gate at the top=20 and plus gates at the bottom, i.e the circuit computes a=20 symmetric function in linear functions -$S_{m}^{d}(\ell_1,\ell_2,...,\ell_m)$($S_{m}^{d}$is the$d$'th=20 elementary symmetric polynomial in$m$... more >>> TR04-028 | 19th March 2004 Arfst Nickelsen, Till Tantau, Lorenz Weizsäcker #### Aggregates with Component Size One Characterize Polynomial Space Aggregates are a computational model similar to circuits, but the underlying graph is not necessarily acyclic. Logspace-uniform polynomial-size aggregates decide exactly the languages in PSPACE; without uniformity condition they decide the languages in PSPACE/poly. As a measure of similarity to boolean circuits we introduce the parameter component size. We ... more >>> TR10-185 | 2nd December 2010 Vitaly Feldman, Venkatesan Guruswami, Prasad Raghavendra, Yi Wu #### Agnostic Learning of Monomials by Halfspaces is Hard We prove the following strong hardness result for learning: Given a distribution of labeled examples from the hypercube such that there exists a monomial consistent with$(1-\epsilon)$of the examples, it is$\mathrm{NP}$-hard to find a halfspace that is correct on$(1/2+\epsilon)$of the examples, for arbitrary constants$\epsilon > ... more >>>

TR94-020 | 12th December 1994

#### Agnostic PAC-Learning of Functions on Analog Neural Nets

We consider learning on multi-layer neural nets with piecewise polynomial
activation functions and a fixed number k of numerical inputs. We exhibit
arbitrarily large network architectures for which efficient and provably
successful learning algorithms exist in the rather realistic refinement of
Valiant's model for probably approximately correct learning ("PAC-learning")
where ... more >>>

TR00-013 | 14th February 2000
Daniel Král

#### Algebraic and Uniqueness Properties of Parity Ordered Binary Decision Diagrams and their Generalization

Ordered binary decision diagrams (OBDDs) and parity ordered binary
decision diagrams have proved their importance as data structures
representing Boolean functions. In addition to parity OBDDs we study
their generalization which we call parity AOBDDs, give the algebraic
characterization theorem and compare their minimal size to the size
more >>>

TR11-022 | 14th February 2011
Malte Beecken, Johannes Mittmann, Nitin Saxena

#### Algebraic Independence and Blackbox Identity Testing

Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. Polynomials $\{f_1,\ldots, f_m\} \subset \mathbb{F}[x_1,\ldots, x_n]$ are called algebraically independent if there is no non-zero polynomial $F$ such that $F(f_1, \ldots, f_m) = 0$. The transcendence degree, $\mbox{trdeg}\{f_1,\ldots, f_m\}$, is the maximal ... more >>>

TR12-014 | 20th February 2012
Johannes Mittmann, Nitin Saxena, Peter Scheiblechner

#### Algebraic Independence in Positive Characteristic -- A p-Adic Calculus

A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the well-known Jacobian criterion. For fields of other characteristic $p>0$, there is no analogous characterization known. In this paper we give the first such criterion. Essentially, it boils down to ... more >>>

TR07-022 | 20th February 2007
Rafail Ostrovsky, William Skeith

#### Algebraic Lower Bounds for Computing on Encrypted Data

In cryptography, there has been tremendous success in building
primitives out of homomorphic semantically-secure encryption
schemes, using homomorphic properties in a black-box way. A few
notable examples of such primitives include items like private
information retrieval schemes and collision-resistant hash functions. In this paper, we illustrate a general
methodology for ... more >>>

TR01-011 | 21st January 2001
Dima Grigoriev, Edward Hirsch

#### Algebraic proof systems over formulas

We introduce two algebraic propositional proof systems F-NS
and F-PC. The main difference of our systems from (customary)
Nullstellensatz and Polynomial Calculus is that the polynomials
are represented as arbitrary formulas (rather than sums of
monomials). Short proofs of Tseitin's tautologies in the
constant-depth version of F-NS provide ... more >>>

TR10-097 | 16th June 2010
Iddo Tzameret

#### Algebraic Proofs over Noncommutative Formulas

Revisions: 1

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege--yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analogue of Frege proofs, different from that given in Buss ... more >>>

TR07-111 | 1st November 2007

#### Algebraic Property Testing: The Role of Invariance

We argue that the symmetries of a property being tested play a
central role in property testing. We support this assertion in the
context of algebraic functions, by examining properties of functions
mapping a vector space $\K^n$ over a field $\K$ to a subfield $\F$.
We consider $\F$-linear properties that ... more >>>

TR05-132 | 8th November 2005
Venkatesan Guruswami

#### Algebraic-geometric generalizations of the Parvaresh-Vardy codes

This paper is concerned with a new family of error-correcting codes
based on algebraic curves over finite fields, and list decoding
algorithms for them. The basic goal in the subject of list decoding is
to construct error-correcting codes $C$ over some alphabet $\Sigma$
which have good rate $R$, and at ... more >>>

TR08-005 | 15th January 2008
Scott Aaronson, Avi Wigderson

#### Algebrization: A New Barrier in Complexity Theory

Any proof of P!=NP will have to overcome two barriers: relativization
and natural proofs. Yet over the last decade, we have seen circuit
lower bounds (for example, that PP does not have linear-size circuits)
that overcome both barriers simultaneously. So the question arises of
whether there ... more >>>

TR08-039 | 7th April 2008
Oded Goldreich, Dana Ron

#### Algorithmic Aspects of Property Testing in the Dense Graphs Model

In this paper we consider two refined questions regarding
the query complexity of testing graph properties
The first question refers to the relation between adaptive
and non-adaptive testers, whereas the second question refers to
testability within complexity that
is inversely proportional to ... more >>>

TR11-128 | 21st September 2011
Michael Elberfeld, Andreas Jakoby, Till Tantau

#### Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth

An algorithmic meta theorem for a logic and a class $C$ of structures states that all problems expressible in this logic can be solved efficiently for inputs from $C$. The prime example is Courcelle's Theorem, which states that monadic second-order (MSO) definable problems are linear-time solvable on graphs of bounded ... more >>>

TR09-147 | 30th December 2009
Stephan Kreutzer

#### Algorithmic Meta-Theorems

Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. They often have a logical and a
structural component, that is they are results of the form:
"every computational problem that can be formalised in a given logic L ... more >>>

TR10-073 | 21st April 2010
Neeraj Kayal

#### Algorithms for Arithmetic Circuits

Given a multivariate polynomial f(x) in F[x] as an arithmetic circuit we would like to efficiently determine:

(i) [Identity Testing.] Is f(x) identically zero?

(ii) [Degree Computation.] Is the degree of the
polynomial f(x) at most a given integer d.

(iii) [Polynomial Equivalence.] Upto an invertible ... more >>>

TR05-033 | 5th March 2005

#### Algorithms for Counting 2-SAT Solutions and Colorings with Applications

An algorithm is presented for counting the number of maximum weight satisfying assignments of a 2SAT formula. The worst case running time of $O(\mbox{poly($n$)} \cdot 1.2461^n)$ for formulas with $n$ variables improves on the previous bound of $O(\mbox{poly($n$)} \cdot 1.2561^n)$ by Dahll\"of, Jonsson, and Wahlstr\"om . The weighted 2SAT counting ... more >>>

TR07-059 | 6th July 2007
Shankar Kalyanaraman, Chris Umans

#### Algorithms for Playing Games with Limited Randomness

only limited randomness. This constrains both the algorithms used to
compute equilibria (they should use little or no randomness) as well
as the mixed strategies that the participants are capable of playing
(these should be sparse). We frame algorithmic ... more >>>

TR01-012 | 4th January 2001
Evgeny Dantsin, Edward Hirsch, Sergei Ivanov, Maxim Vsemirnov

#### Algorithms for SAT and Upper Bounds on Their Complexity

We survey recent algorithms for the propositional
satisfiability problem, in particular algorithms
that have the best current worst-case upper bounds
on their complexity. We also discuss some related
issues: the derandomization of the algorithm of
Paturi, Pudlak, Saks and Zane, the Valiant-Vazirani
Lemma, and random walk ... more >>>

TR03-072 | 15th September 2003
Evgeny Dantsin, Edward Hirsch, Alexander Wolpert

#### Algorithms for SAT based on search in Hamming balls

We present a simple randomized algorithm for SAT and prove an upper
bound on its running time. Given a Boolean formula F in conjunctive
normal form, the algorithm finds a satisfying assignment for F
(if any) by repeating the following: Choose an assignment A at
random and ... more >>>

TR10-122 | 18th July 2010
Zhixiang Chen, Bin Fu, Yang Liu, Robert Schweller

#### Algorithms for Testing Monomials in Multivariate Polynomials

This paper is our second step towards developing a theory of
testing monomials in multivariate polynomials. The central
question is to ask whether a polynomial represented by an
arithmetic circuit has some types of monomials in its sum-product
expansion. The complexity aspects of this problem and its variants
have been ... more >>>

TR11-124 | 15th September 2011

#### Algorithms for the Coin Weighing Problems with the Presence of Noise

The coin weighing problem is the following: Given $n$ coins for which $m$ of them are counterfeit with the same weight. The problem is to detect the counterfeit coins with minimal number of weighings. This problem has many applications in compressed sensing, multiple access adder channels, etc. The problem was ... more >>>

TR06-122 | 20th September 2006
Noam Livne

#### All Natural NPC Problems Have Average-Case Complete Versions

Revisions: 1

In 1984 Levin put forward a suggestion for a theory of {\em average
case complexity}. In this theory a problem, called a {\em
distributional problem}, is defined as a pair consisting of a
decision problem and a probability distribution over the instances.
Introducing adequate notions of simple distributions and average
more >>>

TR97-040 | 17th September 1997
Dorit Dor, Shay Halperin, Uri Zwick

#### All Pairs Almost Shortest Paths

Let G=(V,E) be an unweighted undirected graph on n vertices. A simple
argument shows that computing all distances in G with an additive
one-sided error of at most 1 is as hard as Boolean matrix
multiplication. Building on recent work of Aingworth, Chekuri and
Motwani, we describe an \tilde{O}(min{n^{3/2}m^{1/2},n^{7/3}) time
more >>>

TR00-060 | 17th August 2000
Uri Zwick

#### All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication

We present two new algorithms for solving the {\em All
Pairs Shortest Paths\/} (APSP) problem for weighted directed
graphs. Both algorithms use fast matrix multiplication algorithms.

The first algorithm
solves the APSP problem for weighted directed graphs in which the edge
weights are integers of small absolute value in ... more >>>

TR99-004 | 3rd February 1999
Valentine Kabanets

#### Almost $k$-Wise Independence and Boolean Functions Hard for Read-Once Branching Programs

Revisions: 1

Andreev et al.~\cite{ABCR97} give constructions of Boolean
functions (computable by polynomial-size circuits) that require large
read-once branching program (1-b.p.'s): a function in P that requires
1-b.p. of size at least $2^{n-\polylog(n)}$, a function in quasipolynomial
time that requires 1-b.p. of size at least $2^{n-O(\log n)}$, and a
function in LINSPACE ... more >>>

TR02-048 | 31st July 2002
Noga Alon, Oded Goldreich, Yishay Mansour

#### Almost $k$-wise independence versus $k$-wise independence

We say that a distribution over $\{0,1\}^n$
is almost $k$-wise independent
if its restriction to every $k$ coordinates results in a
distribution that is close to the uniform distribution.
A natural question regarding almost $k$-wise independent
distributions is how close they are to some $k$-wise
independent distribution. We show that ... more >>>

TR05-010 | 8th December 2004
Olivier Powell

#### Almost Completeness in Small Complexity Classes

We constructively prove the existence of almost complete problems under logspace manyone reduction for some small complexity classes by exhibiting a parametrizable construction which yields, when appropriately setting the parameters, an almost complete problem for PSPACE, the class of space efficiently decidable problems, and for SUBEXP, the class of problems ... more >>>

TR07-012 | 22nd January 2007
Shachar Lovett, Sasha Sodin

#### Almost Euclidean sections of the N-dimensional cross-polytope using O(N) random bits

Revisions: 1

It is well known that $\R^N$ has subspaces of dimension
proportional to $N$ on which the $\ell_1$ norm is equivalent to the
$\ell_2$ norm; however, no explicit constructions are known.
Extending earlier work by Artstein--Avidan and Milman, we prove that
such a subspace can be generated using $O(N)$ random bits.

... more >>>

TR07-086 | 7th September 2007
Venkatesan Guruswami, James R. Lee, Alexander Razborov

#### Almost Euclidean subspaces of $\ell_1^N$ via expander codes

We give an explicit (in particular, deterministic polynomial time)
construction of subspaces $X \subseteq \R^N$ of dimension $(1-o(1))N$ such that for every $x \in X$,
$$(\log N)^{-O(\log\log\log N)} \sqrt{N}\, \|x\|_2 \leq \|x\|_1 \leq \sqrt{N}\, \|x\|_2.$$
If we are allowed to use $N^{1/\log\log N}\leq N^{o(1)}$ random bits
and ... more >>>

TR11-049 | 9th April 2011
Noga Alon, Shachar Lovett

#### Almost k-wise vs. k-wise independent permutations, and uniformity for general group actions

A family of permutations in $S_n$ is $k$-wise independent if a uniform permutation chosen from the family maps any distinct $k$ elements to any distinct $k$ elements equally likely. Efficient constructions of $k$-wise independent permutations are known for $k=2$ and $k=3$, but are unknown for $k \ge 4$. In fact, ... more >>>

TR09-120 | 18th November 2009
Charanjit Jutla

#### Almost Optimal Bounds for Direct Product Threshold Theorem

Revisions: 2

We consider weakly-verifiable puzzles which are challenge-response puzzles such that the responder may not
be able to verify for itself whether it answered the challenge correctly. We consider $k$-wise direct product of
such puzzles, where now the responder has to solve $k$ puzzles chosen independently in parallel.
Canetti et ... more >>>

TR10-183 | 29th November 2010
Raghu Meka

#### Almost Optimal Explicit Johnson-Lindenstrauss Transformations

Revisions: 2

The Johnson-Lindenstrauss lemma is a fundamental result in probability with several applications in the design and analysis of algorithms in high dimensional geometry. Most known constructions of linear embeddings that satisfy the Johnson-Lindenstrauss property involve randomness. We address the question of explicitly constructing such embedding families and provide a construction ... more >>>

TR03-066 | 2nd September 2003
Daniele Micciancio

#### Almost perfect lattices, the covering radius problem, and applications to Ajtai's connection factor

Lattices have received considerable attention as a potential source of computational hardness to be used in cryptography, after a breakthrough result of Ajtai (STOC 1996) connecting the average-case and worst-case complexity of various lattice problems. The purpose of this paper is twofold. On the expository side, we present a rigorous ... more >>>

TR08-038 | 4th April 2008
Eric Allender, Michal Koucky

#### Amplifying Lower Bounds by Means of Self-Reducibility

Revisions: 2

We observe that many important computational problems in NC^1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC^0 circuits if and only if it has TC^0 circuits of size n^{1+\epsilon} for every \epsilon > 0 (counting the ... more >>>

TR11-157 | 25th November 2011
Eli Ben-Sasson, Shachar Lovett, Noga Zewi

#### An additive combinatorics approach to the log-rank conjecture in communication complexity

Revisions: 1

For a {0,1}-valued matrix $M$ let CC($M$) denote the deterministic communication complexity of the boolean function associated with $M$. The log-rank conjecture of Lovasz and Saks [FOCS 1988] states that CC($M$) is at most $\log^c({\mbox{rank}}(M))$ for some absolute constant $c$ where rank($M$) denotes the rank of $M$ over the field ... more >>>

TR96-010 | 9th February 1996
Christoph Meinel, Anna Slobodova

#### An Adequate Reducibility Concept for Problems Defined in Terms of Ordered Binary Decision Diagrams

Revisions: 1

Reducibility concepts are fundamental in complexity theory.
Usually, they are defined as follows: A problem P is reducible
to a problem S if P can be computed using a program or device
for S as a subroutine. However, in the case of such restricted
models as ... more >>>

TR11-172 | 20th December 2011
Yang Cai, Constantinos Daskalakis, S. Matthew Weinberg

#### An Algorithmic Characterization of Multi-Dimensional Mechanisms

We obtain a characterization of feasible, Bayesian, multi-item multi-bidder mechanisms with independent, additive bidders as distributions over hierarchical mechanisms. Combined with cyclic-monotonicity our results provide a complete characterization of feasible, Bayesian Incentive Compatible mechanisms for this setting.

Our characterization is enabled by a novel, constructive proof of Border's theorem [Border ... more >>>

TR08-108 | 19th November 2008

#### An Almost Optimal Rank Bound for Depth-3 Identities

We show that the rank of a depth-3 circuit (over any field) that is simple,
minimal and zero is at most O(k^3\log d). The previous best rank bound known was
2^{O(k^2)}(\log d)^{k-2} by Dvir and Shpilka (STOC 2005).
This almost resolves the rank question first posed by ... more >>>

TR10-096 | 16th June 2010
Dana Moshkovitz

#### An Alternative Proof of The Schwartz-Zippel Lemma

Revisions: 1

We show a non-inductive proof of the Schwartz-Zippel lemma. The lemma bounds the number of zeros of a multivariate low degree polynomial over a finite field.

more >>>

TR00-018 | 16th February 2000
Oliver Kullmann

#### An application of matroid theory to the SAT problem

A basic property of minimally unsatisfiable clause-sets F is that
c(F) >= n(F) + 1 holds, where c(F) is the number of clauses, and
n(F) the number of variables. Let MUSAT(k) be the class of minimally
unsatisfiable clause-sets F with c(F) <= n(F) + k.

Poly-time decision algorithms are known ... more >>>

TR04-110 | 24th November 2004
Tomoyuki Yamakami, Harumichi Nishimura

#### An Application of Quantum Finite Automata to Interactive Proof Systems

Quantum finite automata have been studied intensively since
their introduction in late 1990s as a natural model of a
quantum computer with finite-dimensional quantum memory space.
This paper seeks their direct application
to interactive proof systems in which a mighty quantum prover
communicates with a ... more >>>

TR09-085 | 14th September 2009
Christoph Behle, Andreas Krebs, Stephanie Reifferscheid

#### An Approach to characterize the Regular Languages in TC0 with Linear Wires

Revisions: 1

We consider the regular languages recognized by weighted threshold circuits with a linear number of wires.
We present a simple proof to show that parity cannot be computed by such circuits.
Our proofs are based on an explicit construction to restrict the input of the circuit such that the value ... more >>>

TR97-017 | 5th May 1997
Marek Karpinski, Juergen Wirtgen, Alexander Zelikovsky

#### An Approximation Algorithm for the Bandwidth Problem on Dense Graphs

The bandwidth problem is the problem of numbering the vertices of a
given graph $G$ such that the maximum difference between the numbers
of adjacent vertices is minimal. The problem has a long history and
is known to be NP-complete Papadimitriou [Pa76]. Only few special
cases ... more >>>

TR04-048 | 21st April 2004
André Lanka, Andreas Goerdt

#### An approximation hardness result for bipartite Clique

Assuming 3-SAT formulas are hard to refute
on average, Feige showed some approximation hardness
results for several problems like min bisection, dense
$k$-subgraph, max bipartite clique and the 2-catalog segmentation
problem. We show a similar result for
max bipartite clique, but under the assumption, 4-SAT formulas
are hard to refute ... more >>>

TR98-069 | 7th December 1998
Rüdiger Reischuk, Thomas Zeugmann

#### An Average-Case Optimal One-Variable Pattern Language Learner

A new algorithm for learning one-variable pattern languages from positive data
is proposed and analyzed with respect to its average-case behavior.
We consider the total learning time that takes into account all
operations till convergence to a correct hypothesis is achieved.

For almost all meaningful distributions
defining how the ... more >>>

TR95-038 | 2nd July 1995
Joe Kilian, Erez Petrank

#### An Efficient Non-Interactive Zero-Knowledge Proof System for NP with General Assumptions

We consider noninteractive zero-knowledge proofs in the shared random
string model proposed by Blum, Feldman and Micali \cite{bfm}. Until
recently there was a sizable polynomial gap between the most
efficient noninteractive proofs for {\sf NP} based on general
complexity assumptions \cite{fls} versus those based on specific
algebraic assumptions \cite{Da}. ... more >>>

TR06-119 | 13th September 2006
Noga Alon, Oded Schwartz, Asaf Shapira

#### An Elementary Construction of Constant-Degree Expanders

We describe a short and easy to analyze construction of
constant-degree expanders. The construction relies on the
replacement-product, which we analyze using an elementary
combinatorial argument. The construction applies the replacement
product (only twice!) to turn the Cayley expanders of \cite{AR},
whose degree is polylog n, into constant degree
expanders.

... more >>>

TR11-026 | 27th February 2011
Evgeny Demenkov, Alexander Kulikov

#### An Elementary Proof of $3n-o(n)$ Lower Bound on the Circuit Complexity of Affine Dispersers

A Boolean function $f \colon \mathbb{F}^n_2 \rightarrow \mathbb{F}_2$ is called an affine disperser for sources of dimension $d$, if $f$ is not constant on any affine subspace of $\mathbb{F}^n_2$ of dimension at least $d$. Recently Ben-Sasson and Kopparty gave an explicit construction of an affine disperser for $d=o(n)$. The main ... more >>>

TR10-182 | 26th November 2010
Shachar Lovett

#### An elementary proof of anti-concentration of polynomials in Gaussian variables

Recently there has been much interest in polynomial threshold functions in the context of learning theory, structural results and pseudorandomness. A crucial ingredient in these works is the understanding of the distribution of low-degree multivariate polynomials evaluated over normally distributed inputs. In particular, the two important properties are exponential tail ... more >>>

TR10-091 | 14th May 2010
Nikolay Vereshchagin

#### An Encoding Invariant Version of Polynomial Time Computable Distributions

When we represent a decision problem,like CIRCUIT-SAT, as a language over the binary alphabet,
we usually do not specify how to encode instances by binary strings.
This relies on the empirical observation that the truth of a statement of the form CIRCUIT-SAT belongs to a complexity class $C$''
more >>>

TR01-083 | 29th October 2001
Nikolay Vereshchagin

#### An enumerable undecidable set with low prefix complexity: a simplified proof

Revisions: 1

We present a simplified proof of Solovay-Calude-Coles theorem
stating that there is an enumerable undecidable set with the
following property: prefix
complexity of its initial segment of length n is bounded by prefix
complexity of n (up to an additive constant).

more >>>

TR03-013 | 7th March 2003
Luca Trevisan

#### An epsilon-Biased Generator in NC0

Cryan and Miltersen recently considered the question
of whether there can be a pseudorandom generator in
NC0, that is, a pseudorandom generator such that every
bit of the output depends on a constant number k of bits
of the seed. They show that for k=3 there ... more >>>

TR12-127 | 3rd October 2012

#### An Explicit VC-Theorem for Low-Degree Polynomials

Let $X \subseteq \mathbb{R}^{n}$ and let ${\mathcal C}$ be a class of functions mapping $\mathbb{R}^{n} \rightarrow \{-1,1\}.$ The famous VC-Theorem states that a random subset $S$ of $X$ of size $O(\frac{d}{\epsilon^{2}} \log \frac{d}{\epsilon})$, where $d$ is the VC-Dimension of ${\mathcal C}$, is (with constant probability) an $\epsilon$-approximation for ${\mathcal C}$ ... more >>>

TR07-007 | 17th January 2007
Jan Krajicek

#### An exponential lower bound for a constraint propagation proof system based on ordered binary decision diagrams

We prove an exponential lower bound on the size of proofs
in the proof system operating with ordered binary decision diagrams
introduced by Atserias, Kolaitis and Vardi. In fact, the lower bound
applies to semantic derivations operating with sets defined by OBDDs.
We do not assume ... more >>>

TR12-098 | 3rd August 2012
Ankit Gupta, Pritish Kamath, Neeraj Kayal, Ramprasad Saptharishi

#### An exponential lower bound for homogeneous depth four arithmetic circuits with bounded bottom fanin

Revisions: 2

Agrawal and Vinay (FOCS 2008) have recently shown that an exponential lower bound for depth four homogeneous circuits with bottom layer of $\times$ gates having sublinear fanin translates to an exponential lower bound for a general arithmetic circuit computing the permanent. Motivated by this, we examine the complexity of computing ... more >>>

TR12-081 | 26th June 2012
Neeraj Kayal

#### An exponential lower bound for the sum of powers of bounded degree polynomials

Revisions: 1

In this work we consider representations of multivariate polynomials in $F[x]$ of the form $f(x) = Q_1(x)^{e_1} + Q_2(x)^{e_2} + ... + Q_s(x)^{e_s},$ where the $e_i$'s are positive integers and the $Q_i$'s are arbitary multivariate polynomials of bounded degree. We give an explicit $n$-variate polynomial $f$ of degree $n$ ... more >>>

TR95-057 | 24th November 1995
Dima Grigoriev, Marek Karpinski, A. C. Yao

#### An Exponential Lower Bound on the Size of Algebraic Decision Trees for MAX

We prove an exponential lower bound on the size of any
fixed-degree algebraic decision tree for solving MAX, the
problem of finding the maximum of $n$ real numbers. This
complements the $n-1$ lower bound of Rabin \cite{R72} on
the depth of algebraic ... more >>>

TR94-018 | 12th December 1994
Jan Krajicek, Pavel Pudlak, Alan Woods

#### An Exponential Lower Bound to the Size of Bounded Depth Frege Proofs of the Pigeonhole Principle

We prove lower bounds of the form $exp\left(n^{\epsilon_d}\right),$
$\epsilon_d>0,$ on the length of proofs of an explicit sequence of
tautologies, based on the Pigeonhole Principle, in proof systems
using formulas of depth $d,$ for any constant $d.$ This is the
largest lower bound for the strongest proof system, for which ... more >>>

TR01-056 | 6th August 2001
Michael Alekhnovich, Jan Johannsen, Alasdair Urquhart

#### An Exponential Separation between Regular and General Resolution

This paper gives two distinct proofs of an exponential separation
between regular resolution and unrestricted resolution.
The previous best known separation between these systems was
quasi-polynomial.

more >>>

TR07-046 | 23rd April 2007
Philipp Hertel

#### An Exponential Time/Space Speedup For Resolution

Satisfiability algorithms have become one of the most practical and successful approaches for solving a variety of real-world problems, including hardware verification, experimental design, planning and diagnosis problems. The main reason for the success is due to highly optimized algorithms for SAT based on resolution. The most successful of these ... more >>>

TR07-034 | 29th March 2007
Anup Rao

#### An Exposition of Bourgain's 2-Source Extractor

A construction of Bourgain gave the first 2-source
extractor to break the min-entropy rate 1/2 barrier. In this note,
we write an exposition of his result, giving a high level way to view
his extractor construction.

We also include a proof of a generalization of Vazirani's XOR lemma
that seems ... more >>>

TR12-029 | 3rd April 2012
Shachar Lovett

#### An exposition of Sanders quasi-polynomial Freiman-Ruzsa theorem

The polynomial Freiman-Ruzsa conjecture is one of the important conjectures in additive combinatorics. It asserts than one can switch between combinatorial and algebraic notions of approximate subgroups with only a polynomial loss in the underlying parameters. This conjecture has also already found several applications in theoretical computer science. Recently, Tom ... more >>>

TR05-067 | 28th June 2005
Zeev Dvir, Amir Shpilka

#### An Improved Analysis of Mergers

Mergers are functions that transform k (possibly dependent) random sources into a single random source, in a way that ensures that if one of the input sources has min-entropy rate $\delta$ then the output has min-entropy rate close to $\delta$. Mergers have proven to be a very useful tool in ... more >>>

TR06-107 | 26th August 2006

#### An improved bound on correlation between polynomials over Z_m and MOD_q

Revisions: 1

Let m,q > 1 be two integers that are co-prime and A be any subset of Z_m. Let P be any multi-linear polynomial of degree d in n variables over Z_m. We show that the MOD_q boolean function on n variables has correlation at most exp(-\Omega(n/(m2^{m-1})^d)) with the boolean function ... more >>>

TR00-057 | 25th July 2000
Martin Sauerhoff

#### An Improved Hierarchy Result for Partitioned BDDs

One of the great challenges of complexity theory is the problem of
analyzing the dependence of the complexity of Boolean functions on the
resources nondeterminism and randomness. So far, this problem could be
solved only for very few models of computation. For so-called
partitioned binary decision diagrams, which are a ... more >>>

TR12-099 | 5th August 2012
Nikos Leonardos

#### An improved lower bound for the randomized decision tree complexity of recursive majority

Revisions: 1

We prove that the randomized decision tree complexity of the recursive majority-of-three is $\Omega(2.6^d)$, where $d$ is the depth of the recursion. The proof is by a bottom up induction, which is same in spirit as the one in the proof of Saks and Wigderson in their FOCS 1986 paper ... more >>>

TR05-030 | 12th February 2005
Evgeny Dantsin, Alexander Wolpert

#### An Improved Upper Bound for SAT

We give a randomized algorithm for testing satisfiability of Boolean formulas in conjunctive normal form with no restriction on clause length. Its running time is at most $2^{n(1-1/\alpha)}$ up to a polynomial factor, where $\alpha = \ln(m/n) + O(\ln \ln m)$ and $n$, $m$ are respectively the number of variables ... more >>>

TR07-117 | 8th November 2007
Edward Hirsch, Dmitry Itsykson

#### An infinitely-often one-way function based on an average-case assumption

We assume the existence of a function f that is computable in polynomial time but its inverse function is not computable in randomized average-case polynomial time. The cryptographic setting is, however, different: even for a weak one-way function, every possible adversary should fail on a polynomial fraction of inputs. Nevertheless, ... more >>>

TR12-131 | 18th October 2012
Mark Braverman, Ankur Moitra

#### An Information Complexity Approach to Extended Formulations

We prove an unconditional lower bound that any linear program that achieves an $O(n^{1-\epsilon})$ approximation for clique has size $2^{\Omega(n^\epsilon)}$. There has been considerable recent interest in proving unconditional lower bounds against any linear program. Fiorini et al proved that there is no polynomial sized linear program for traveling salesman. ... more >>>

TR09-144 | 24th December 2009

#### An Invariance Principle for Polytopes

Let $X$ be randomly chosen from $\{-1,1\}^n$, and let $Y$ be randomly
chosen from the standard spherical Gaussian on $\R^n$. For any (possibly unbounded) polytope $P$
formed by the intersection of $k$ halfspaces, we prove that
\left|\Pr\left[X \in P\right] - \Pr\left[Y \in P\right]\right| \leq \log^{8/5}k \cdot ... more >>>

TR06-011 | 2nd January 2006
Yijia Chen, Martin Grohe

#### An Isomorphism between Subexponential and Parameterized Complexity Theory

We establish a close connection between (sub)exponential time complexity and parameterized complexity by proving that the so-called miniaturization mapping is a reduction preserving isomorphism between the two theories.

more >>>

TR96-002 | 10th January 1996
Manindra Agrawal, Eric Allender

#### An Isomorphism Theorem for Circuit Complexity

We show that all sets complete for NC$^1$ under AC$^0$
reductions are isomorphic under AC$^0$-computable isomorphisms.

Although our proof does not generalize directly to other
complexity classes, we do show that, for all complexity classes C
closed under NC$^1$-computable many-one reductions, the sets
more >>>

TR04-114 | 21st November 2004

#### An O(log n log log n) Space Algorithm for Undirected s,t-Connectivity

We present a deterministic O(log n log log n) space algorithm for
undirected s,t-connectivity. It is based on the deterministic EREW
algorithm of Chong and Lam (SODA 93) and uses the universal
exploration sequences for trees constructed by Kouck\'y (CCC 01).
Our result improves the O(log^{4/3} n) bound of Armoni ... more >>>

TR06-050 | 18th April 2006
Alexander Razborov, Sergey Yekhanin

#### An Omega(n^{1/3}) Lower Bound for Bilinear Group Based Private Information Retrieval

A two server private information retrieval (PIR) scheme
allows a user U to retrieve the i-th bit of an
n-bit string x replicated between two servers while each
server individually learns no information about i. The main
parameter of interest in a PIR scheme is its communication
complexity, namely the ... more >>>

TR13-062 | 18th April 2013

#### An optimal lower bound for monotonicity testing over hypergrids

For positive integers $n, d$, consider the hypergrid $[n]^d$ with the coordinate-wise product partial ordering denoted by $\prec$.
A function $f: [n]^d \mapsto \mathbb{N}$ is monotone if $\forall x \prec y$, $f(x) \leq f(y)$.
A function $f$ is $\varepsilon$-far from monotone if at least an $\varepsilon$-fraction of values must ... more >>>

TR10-140 | 17th September 2010
Amit Chakrabarti, Oded Regev

#### An Optimal Lower Bound on the Communication Complexity of Gap-Hamming-Distance

We prove an optimal $\Omega(n)$ lower bound on the randomized
communication complexity of the much-studied
Gap-Hamming-Distance problem. As a consequence, we
obtain essentially optimal multi-pass space lower bounds in the
data stream model for a number of fundamental problems, including
the estimation of frequency moments.

The Gap-Hamming-Distance problem is a ... more >>>

TR03-070 | 19th August 2003
Amit Chakrabarti, Oded Regev

#### An Optimal Randomised Cell Probe Lower Bound for Approximate Nearest Neighbour Searching

We consider the approximate nearest neighbour search problem on the
Hamming Cube $\b^d$. We show that a randomised cell probe algorithm that
uses polynomial storage and word size $d^{O(1)}$ requires a worst case
query time of $\Omega(\log\log d/\log\log\log d)$. The approximation
factor may be as loose as $2^{\log^{1-\eta}d}$ for any ... more >>>

TR06-056 | 27th April 2006

#### An Unconditional Study of Computational Zero Knowledge

We prove a number of general theorems about ZK, the class of problems possessing (computational) zero-knowledge proofs. Our results are unconditional, in contrast to most previous works on ZK, which rely on the assumption that one-way functions exist.

We establish several new characterizations of ZK, and use these characterizations to ... more >>>

TR95-010 | 16th February 1995
Pavel Pudlak, Jiri Sgall

#### An Upper Bound for a Communication Game Related to Time-Space Tradeoffs

We prove an unexpected upper bound on a communication game proposed
by Jeff Edmonds and Russell Impagliazzo as an approach for
proving lower bounds for time-space tradeoffs for branching programs.
Our result is based on a generalization of a construction of Erdos,
Frankl and Rodl of a large 3-hypergraph with ... more >>>

TR01-079 | 6th September 2001
Michele Zito

#### An Upper Bound on the Space Complexity of Random Formulae in Resolution

We prove that, with high probability, the space complexity of refuting
a random unsatisfiable boolean formula in $k$-CNF on $n$
variables and $m = \Delta n$ clauses is
$O(n \cdot \Delta^{-\frac{1}{k-2}})$.

more >>>

TR97-052 | 11th November 1997
Eduardo D. Sontag

#### Analog Neural Nets with Gaussian or other Common Noise Distributions cannot Recognize Arbitrary Regular Languages

We consider recurrent analog neural nets where the output of each
gate is subject to Gaussian noise, or any other common noise
distribution that is nonzero on a large set.
We show that many regular languages cannot be recognized by
networks of this type, and
more >>>

TR95-025 | 8th May 1995
Günter Hotz, Gero Vierke, Bjoern Schieffer

#### Analytic Machines

In this paper the $R$-machines defined by Blum, Shub and Smale
are generalized by allowing infinite convergent computations.
The description of real numbers is infinite.
Therefore, considering arithmetic operations on real numbers should
also imply infinite computations on {\em analytic machines}.
We prove that $\R$-computable functions are $\Q$-analytic.
We show ... more >>>

TR05-025 | 20th February 2005
Zeev Dvir, Ran Raz

#### Analyzing Linear Mergers

Mergers are functions that transform k (possibly dependent)
random sources into a single random source, in a way that ensures
that if one of the input sources has min-entropy rate $\delta$
then the output has min-entropy rate close to $\delta$. Mergers
have proven to be a very useful tool in ... more >>>

TR12-022 | 14th March 2012
Amit Chakrabarti, Graham Cormode, Andrew McGregor, Justin Thaler

#### Annotations in Data Streams

The central goal of data stream algorithms is to process massive streams of data using sublinear storage space. Motivated by work in the database community on outsourcing database and data stream processing, we ask whether the space usage of such algorithms can be further reduced by enlisting a more powerful ... more >>>

TR97-045 | 29th September 1997
Oded Goldreich, David Zuckerman

#### Another proof that BPP subseteq PH (and more).

We provide another proof of the Sipser--Lautemann Theorem
by which $BPP\subseteq MA$ ($\subseteq PH$).
The current proof is based on strong
results regarding the amplification of $BPP$, due to Zuckerman.
Given these results, the current proof is even simpler than previous ones.
Furthermore, extending the proof leads to ... more >>>

TR06-143 | 15th November 2006
Frank Neumann, Carsten Witt

#### Ant Colony Optimization and the Minimum Spanning Tree Problem

Ant Colony Optimization (ACO) is a kind of randomized search heuristic that has become very popular for solving problems from combinatorial optimization. Solutions for a given problem are constructed by a random walk on a so-called construction graph. This random walk can be influenced by heuristic information about the problem. ... more >>>

TR00-052 | 3rd July 2000
Beate Bollig, Ingo Wegener

#### Approximability and Nonapproximability by Binary Decision Diagrams

Many BDD (binary decision diagram) models are motivated
by CAD applications and have led to complexity theoretical
problems and results. Motivated by applications in genetic
programming Krause, Savick\'y, and Wegener (1999) have shown
that for the inner product function IP$_n$ and the direct
storage access function DSA$_n$ ... more >>>

TR00-091 | 21st December 2000
Cristina Bazgan, Wenceslas Fernandez de la Vega, Marek Karpinski

#### Approximability of Dense Instances of NEAREST CODEWORD Problem

We give a polynomial time approximation scheme (PTAS) for dense
instances of the NEAREST CODEWORD problem.

more >>>

TR03-056 | 29th July 2003
Piotr Berman, Marek Karpinski

#### Approximability of Hypergraph Minimum Bisection

We prove that the problems of minimum bisection on k-uniform
hypergraphs are almost exactly as hard to approximate,
up to the factor k/3, as the problem of minimum bisection
on graphs. On a positive side, our argument gives also the
first approximation ... more >>>

TR06-045 | 13th March 2006
Jan Arpe, Bodo Manthey

#### Approximability of Minimum AND-Circuits

Revisions: 1

Given a set of monomials, the Minimum AND-Circuit problem asks for a
circuit that computes these monomials using AND-gates of fan-in two and
being of minimum size. We prove that the problem is not polynomial time
approximable within a factor of less than 1.0051 unless P = NP, even if
more >>>

TR02-031 | 30th April 2002
Vikraman Arvind, Venkatesh Raman

#### Approximate Counting small subgraphs of bounded treewidth and related problems

Revisions: 1

We give a randomized approximation algorithm taking
$O(k^{O(k)}n^{b+O(1)})$ time to count the number of copies of a
$k$-vertex graph with treewidth at most $b$ in an $n$ vertex graph
$G$ with approximation ratio $1/k^{O(k)}$ and error probability
inverse exponential in $n$. This algorithm is based on ... more >>>

TR12-078 | 14th June 2012
Vikraman Arvind, Sebastian Kuhnert, Johannes Köbler, Yadu Vasudev

#### Approximate Graph Isomorphism

We study optimization versions of Graph Isomorphism. Given two graphs $G_1,G_2$, we are interested in finding a bijection $\pi$ from $V(G_1)$ to $V(G_2)$ that maximizes the number of matches (edges mapped to edges or non-edges mapped to non-edges). We give an $n^{O(\log n)}$ time approximation scheme that for any constant ... more >>>

TR07-116 | 25th September 2007
Alexander A. Sherstov

#### Approximate Inclusion-Exclusion for Arbitrary Symmetric Functions

Let A_1,...,A_n be events in a probability space. The
approximate inclusion-exclusion problem, due to Linial and
Nisan (1990), is to estimate Pr[A_1 OR ... OR A_n] given
Pr[AND_{i\in S}A_i] for all |S|<=k. Kahn et al. (1996) solve
this problem optimally for each k. We study the following more
general question: ... more >>>

TR10-032 | 19th January 2010

#### Approximate Self-Assembly of the Sierpinski Triangle

The Tile Assembly Model is a Turing universal model that Winfree introduced in order to study the nanoscale self-assembly of complex (typically aperiodic) DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant of the Cartesian plane with specially labeled tiles appearing at exactly the positions of points in ... more >>>

TR11-171 | 15th December 2011
Piotr Indyk, Reut Levi, Ronitt Rubinfeld

#### Approximating and Testing $k$-Histogram Distributions in Sub-linear time

A discrete distribution $p$, over $[n]$, is a $k$-histogram if its probability distribution function can be
represented as a piece-wise constant function with $k$ pieces. Such a function
is
represented by a list of $k$ intervals and $k$ corresponding values. We consider
the following problem: given a collection of samples ... more >>>

TR05-073 | 14th July 2005
Oded Goldreich, Dana Ron

#### Approximating Average Parameters of Graphs.

Inspired by Feige ({\em 36th STOC}, 2004),
we initiate a study of sublinear randomized algorithms
for approximating average parameters of a graph.
Specifically, we consider the average degree of a graph
and the average distance between pairs of vertices in a graph.
Since our focus is on sublinear algorithms, these ... more >>>

TR13-051 | 2nd April 2013
Eric Blais, Li-Yang Tan

#### Approximating Boolean functions with depth-2 circuits

We study the complexity of approximating Boolean functions with DNFs and other depth-2 circuits, exploring two main directions: universal bounds on the approximability of all Boolean functions, and the approximability of the parity function.
In the first direction, our main positive results are the first non-trivial universal upper bounds on ... more >>>

TR01-042 | 31st May 2001
Marek Karpinski

#### Approximating Bounded Degree Instances of NP-Hard Problems

We present some of the recent results on computational complexity
of approximating bounded degree combinatorial optimization problems. In
particular, we present the best up to now known explicit nonapproximability
bounds on the very small degree optimization problems which are of
particular importance on the intermediate stages ... more >>>

TR12-074 | 12th June 2012
Venkatesan Guruswami, Yuan Zhou

#### Approximating Bounded Occurrence Ordering CSPs

A theorem of Håstad shows that for every constraint satisfaction problem (CSP) over a fixed size domain, instances where each variable appears in at most $O(1)$ constraints admit a non-trivial approximation algorithm, in the sense that one can beat (by an additive constant) the approximation ratio achieved by the naive ... more >>>

TR06-007 | 23rd November 2005

#### Approximating Buy-at-Bulk $k$-Steiner trees

In the buy-at-bulk $k$-Steiner tree (or rent-or-buy
$k$-Steiner tree) problem we are given a graph $G(V,E)$ with a set
of terminals $T\subseteq V$ including a particular vertex $s$ called
the root, and an integer $k\leq |T|$. There are two cost functions
on the edges of $G$, a buy cost $b:E\longrightarrow ... more >>> TR07-027 | 2nd February 2007 Tobias Friedrich, Jun He, Nils Hebbinghaus, Frank Neumann, Carsten Witt #### Approximating Covering Problems by Randomized Search Heuristics Using Multi-Objective Models The main aim of randomized search heuristics is to produce good approximations of optimal solutions within a small amount of time. In contrast to numerous experimental results, there are only a few theoretical results on this subject. We consider the approximation ability of randomized search for the class of ... more >>> TR98-048 | 6th July 1998 Irit Dinur, Guy Kindler, Shmuel Safra #### Approximating CVP to Within Almost Polynomial Factor is NP-Hard This paper shows finding the closest vector in a lattice to be NP-hard to approximate to within any factor up to$2^{(\log{n})^{1-\epsilon}}$where$\epsilon = (\log\log{n})^{-\alpha}$and$\alpha$is any positive constant$<{1\over 2}$. more >>> TR97-004 | 19th February 1997 Marek Karpinski, Alexander Zelikovsky #### Approximating Dense Cases of Covering Problems Comments: 1 We study dense instances of several covering problems. An instance of the set cover problem with$m$sets is dense if there is$\epsilon>0$such that any element belongs to at least$\epsilon m$sets. We show that the dense set cover problem can be approximated with ... more >>> TR02-018 | 22nd March 2002 Piotr Berman, Marek Karpinski, Yakov Nekrich #### Approximating Huffman Codes in Parallel In this paper we present some new results on the approximate parallel construction of Huffman codes. Our algorithm achieves linear work and logarithmic time, provided that the initial set of elements is sorted. This is the first parallel algorithm for that problem with the optimal time and ... more >>> TR00-072 | 14th July 2000 Peter Auer, Philip M. Long, Aravind Srinivasan #### Approximating Hyper-Rectangles: Learning and Pseudo-random Sets The PAC learning of rectangles has been studied because they have been found experimentally to yield excellent hypotheses for several applied learning problems. Also, pseudorandom sets for rectangles have been actively studied recently because (i) they are a subproblem common to the derandomization of depth-2 (DNF) ... more >>> TR10-132 | 18th August 2010 Mahdi Cheraghchi, Johan Hastad, Marcus Isaksson, Ola Svensson #### Approximating Linear Threshold Predicates We study constraint satisfaction problems on the domain$\{-1,1\}$, where the given constraints are homogeneous linear threshold predicates. That is, predicates of the form$\mathrm{sgn}(w_1 x_1 + \cdots + w_n x_n)$for some positive integer weights$w_1, \dots, w_n$. Despite their simplicity, current techniques fall short of providing a classification ... more >>> TR03-032 | 16th April 2003 Andreas Björklund, Thore Husfeldt, Sanjeev Khanna #### Approximating Longest Directed Path We investigate the hardness of approximating the longest path and the longest cycle in directed graphs on$n$vertices. We show that neither of these two problems can be polynomial time approximated within$n^{1-\epsilon}$for any$\epsilon>0$unless$\text{P}=\text{NP}$. In particular, the result holds for ... more >>> TR01-025 | 28th March 2001 Piotr Berman, Marek Karpinski #### Approximating Minimum Unsatisfiability of Linear Equations We consider the following optimization problem: given a system of m linear equations in n variables over a certain field, a feasible solution is any assignment of values to the variables, and the minimized objective function is the number of equations that are not satisfied. For ... more >>> TR10-124 | 18th July 2010 Zhixiang Chen, Bin Fu #### Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum multilinear monomial when the input is a$\Pi\Sigma\Pi$polynomial. We first prove ... more >>> TR01-038 | 14th May 2001 Andreas Jakoby, Maciej Liskiewicz, Rüdiger Reischuk #### Approximating Schedules for Dynamic Graphs Efficiently A model for parallel and distributed programs, the dynamic process graph (DPG), is investigated under graph-theoretic and complexity aspects. Such graphs embed constructors for parallel programs, synchronization mechanisms as well as conditional branches. They are capable of representing all possible executions of a parallel or distributed program ... more >>> TR99-002 | 22nd January 1999 Oded Goldreich, Daniele Micciancio, Shmuel Safra and Jean-Pierre Seifert. #### Approximating shortest lattice vectors is not harder than approximating closest lattice vectors. We show that given oracle access to a subroutine which returns approximate closest vectors in a lattice, one may find in polynomial-time approximate shortest vectors in a lattice. The level of approximation is maintained; that is, for any function$f$, the following holds: Suppose that the subroutine, on input a ... more >>> TR99-016 | 25th April 1999 Irit Dinur #### Approximating SVP_\infty to within Almost-Polynomial Factors is NP-hard This paper shows SVP_\infty and CVP_\infty to be NP-hard to approximate to within any factor up to$n^{1/\log\log n}$. This improves on the best previous result \cite{ABSS} that showed quasi-NP-hardness for smaller factors, namely$2^{\log^{1-\epsilon}n}$for any constant$\epsilon>0$. We show a direct reduction from SAT to these problems, that ... more >>> TR13-023 | 6th February 2013 Alexander A. Sherstov #### Approximating the AND-OR Tree The approximate degree of a Boolean function$f$is the least degree of a real polynomial that approximates$f$within$1/3$at every point. We prove that the function$\bigwedge_{i=1}^{n}\bigvee_{j=1}^{n}x_{ij}$, known as the AND-OR tree, has approximate degree$\Omega(n).$This lower bound is tight and closes a ... more >>> TR05-084 | 31st July 2005 Mickey Brautbar, Alex Samorodnitsky #### Approximating the entropy of large alphabets We consider the problem of approximating the entropy of a discrete distribution P on a domain of size q, given access to n independent samples from the distribution. It is known that n > q is necessary, in general, for a good additive estimate of the entropy. A problem of ... more >>> TR12-025 | 23rd March 2012 Kord Eickmeyer, Kristoffer Arnsfelt Hansen, Elad Verbin #### Approximating the minmax value of 3-player games within a constant is as hard as detecting planted cliques We consider the problem of approximating the minmax value of a multiplayer game in strategic form. We argue that in 3-player games with 0-1 payoffs, approximating the minmax value within an additive constant smaller than$\xi/2$, where$\xi = \frac{3-\sqrt5}{2} \approx 0.382$, is not possible by a polynomial time algorithm. ... more >>> TR07-092 | 10th July 2007 Piotr Berman, Bhaskar DasGupta #### Approximating the Online Set Multicover Problems Via Randomized Winnowing In this paper, we consider the weighted online set k-multicover problem. In this problem, we have an universe V of elements, a family SS of subsets of V with a positive real cost for every S\in SS, and a coverage factor'' (positive integer) k. A subset \{i_0,i_1,\ldots\ \subseteq V of ... more >>> TR97-059 | 22nd December 1997 Jin-Yi Cai, Ajay Nerurkar #### Approximating the SVP to within a factor$\left(1 + \frac{1}{\mathrm{dim}^\epsilon}\right)$is NP-hard under randomized reductions Recently Ajtai showed that to approximate the shortest lattice vector in the$l_2$-norm within a factor$(1+2^{-\mbox{\tiny dim}^k})$, for a sufficiently large constant$k$, is NP-hard under randomized reductions. We improve this result to show that to approximate a shortest lattice vector within a factor$(1+ \mbox{dim}^{-\epsilon})$, for any$\epsilon>0$, ... more >>> TR07-119 | 5th December 2007 Piotr Berman, Bhaskar DasGupta, Marek Karpinski #### Approximating Transitive Reductions for Directed Networks We consider <i>minimum equivalent digraph</i> (<i>directed network</i>) problem (also known as the <i>strong transitive reduction</i>), its maximum optimization variant, and some extensions of those two types of problems. We prove the existence of polynomial time approximation algorithms with ratios 1.5 for all the minimization problems and 2 for all the ... more >>> TR06-063 | 1st May 2006 Moses Charikar, Konstantin Makarychev, Yury Makarychev #### Approximation Algorithm for the Max k-CSP Problem We present a c.k/2^k approximation algorithm for the Max k-CSP problem (where c > 0.44 is an absolute constant). This result improves the previously best known algorithm by Hast, which has an approximation guarantee of Omega(k/(2^k log k)). Our approximation guarantee matches the upper bound of Samorodnitsky and Trevisan up ... more >>> TR00-051 | 14th July 2000 Marek Karpinski, Miroslaw Kowaluk, Andrzej Lingas #### Approximation Algorithms for MAX-BISECTION on Low Degree Regular Graphs and Planar Graphs The max-bisection problem is to find a partition of the vertices of a graph into two equal size subsets that maximizes the number of edges with endpoints in both subsets. We obtain new improved approximation ratios for the max-bisection problem on the low degree$k$-regular graphs for ... more >>> TR05-034 | 5th April 2005 Luca Trevisan #### Approximation Algorithms for Unique Games Revisions: 1 , Comments: 1 Khot formulated in 2002 the "Unique Games Conjectures" stating that, for any epsilon > 0, given a system of constraints of a certain form, and the promise that there is an assignment that satisfies a 1-epsilon fraction of constraints, it is intractable to find a solution that satisfies even an ... more >>> TR06-101 | 22nd August 2006 Wenceslas Fernandez de la Vega, Marek Karpinski #### Approximation Complexity of Nondense Instances of MAX-CUT We prove existence of approximation schemes for instances of MAX-CUT with$\Omega(\frac{n^2}{\Delta})$edges which work in$2^{O^\thicksim(\frac{\Delta}{\varepsilon^2})}n^{O(1)}$time. This entails in particular existence of quasi-polynomial approximation schemes (QPTASs) for mildly sparse instances of MAX-CUT with$\Omega(\frac{n^2}{\operatorname{polylog} n})$edges. The result depends on new sampling method for smoothed linear programs that ... more >>> TR96-030 | 31st March 1996 Meera Sitharam #### Approximation from linear spaces and applications to complexity We develop an analytic framework based on linear approximation and point out how a number of complexity related questions -- on circuit and communication complexity lower bounds, as well as pseudorandomness, learnability, and general combinatorics of Boolean functions -- fit neatly into this framework. This ... more >>> TR03-022 | 11th April 2003 Piotr Berman, Marek Karpinski, Alexander D. Scott #### Approximation Hardness and Satisfiability of Bounded Occurrence Instances of SAT We study approximation hardness and satisfiability of bounded occurrence uniform instances of SAT. Among other things, we prove the inapproximability for SAT instances in which every clause has exactly 3 literals and each variable occurs exactly 4 times, and display an explicit ... more >>> TR02-073 | 12th December 2002 Janka Chlebíková, Miroslav Chlebík #### Approximation Hardness for Small Occurrence Instances of NP-Hard Problem The paper contributes to the systematic study (started by Berman and Karpinski) of explicit approximability lower bounds for small occurrence optimization problems. We present parametrized reductions for some packing and covering problems, including 3-Dimensional Matching, and prove the best known inapproximability results even for highly restricted versions of ... more >>> TR01-026 | 3rd April 2001 Piotr Berman, Marek Karpinski #### Approximation Hardness of Bounded Degree MIN-CSP and MIN-BISECTION We consider bounded occurrence (degree) instances of a minimum constraint satisfaction problem MIN-LIN2 and a MIN-BISECTION problem for graphs. MIN-LIN2 is an optimization problem for a given system of linear equations mod 2 to construct a solution that satisfies the minimum number of them. E3-OCC-MIN-E3-LIN2 is ... more >>> TR13-066 | 25th April 2013 Marek Karpinski, Richard Schmied #### Approximation Hardness of Graphic TSP on Cubic Graphs We prove explicit approximation hardness results for the Graphic TSP on cubic and subcubic graphs as well as the new inapproximability bounds for the corresponding instances of the (1,2)-TSP. The proof technique uses new modular constructions of simulating gadgets for the restricted cubic and subcubic instances. The modular constructions used ... more >>> TR03-049 | 25th June 2003 Piotr Berman, Marek Karpinski, Alexander D. Scott #### Approximation Hardness of Short Symmetric Instances of MAX-3SAT We prove approximation hardness of short symmetric instances of MAX-3SAT in which each literal occurs exactly twice, and each clause is exactly of size 3. We display also an explicit approximation lower bound for that problem. The bound two on the number ... more >>> TR00-089 | 1st December 2000 Lars Engebretsen, Marek Karpinski #### Approximation Hardness of TSP with Bounded Metrics Revisions: 1 The general asymmetric (and metric) TSP is known to be approximable only to within an O(log n) factor, and is also known to be approximable within a constant factor as soon as the metric is bounded. In this paper we study the asymmetric and symmetric TSP problems with bounded metrics ... more >>> TR00-058 | 1st August 2000 Martin Sauerhoff #### Approximation of Boolean Functions by Combinatorial Rectangles This paper deals with the number of monochromatic combinatorial rectangles required to approximate a Boolean function on a constant fraction of all inputs, where each rectangle may be defined with respect to its own partition of the input variables. The main result of the paper is that the number of ... more >>> TR06-124 | 25th September 2006 Wenceslas Fernandez de la Vega, Ravi Kannan, Marek Karpinski #### Approximation of Global MAX-CSP Problems We study the problem of absolute approximability of MAX-CSP problems with the global constraints. We prove existence of an efficient sampling method for the MAX-CSP class of problems with linear global constraints and bounded feasibility gap. It gives for the first time a polynomial in epsilon^-1 sample complexity bound for ... more >>> TR12-110 | 4th September 2012 Siu On Chan #### Approximation Resistance from Pairwise Independent Subgroups We show optimal (up to constant factor) NP-hardness for Max-k-CSP over any domain, whenever k is larger than the domain size. This follows from our main result concerning predicates over abelian groups. We show that a predicate is approximation resistant if it contains a subgroup that is ... more >>> TR12-040 | 17th April 2012 Sangxia Huang #### Approximation Resistance on Satisfiable Instances for Predicates Strictly Dominating Parity In this paper, we study the approximability of Max CSP($P$) where$P$is a Boolean predicate. We prove that assuming Khot's$d$-to-1 Conjecture, if the set of accepting inputs of$P$strictly contains all inputs with even (or odd) parity, then it is NP-hard to approximate Max CSP($P$) better than ... more >>> TR08-009 | 7th December 2007 Per Austrin, Elchanan Mossel #### Approximation Resistant Predicates From Pairwise Independence We study the approximability of predicates on$k$variables from a domain$[q]$, and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate$P$is approximation resistant if there exists a balanced pairwise independent distribution over more >>> TR01-065 | 10th August 2001 Chandra Chekuri, Sanjeev Khanna #### Approximation Schemes for Preemptive Weighted Flow Time We present the first approximation schemes for minimizing weighted flow time on a single machine with preemption. Our first result is an algorithm that computes a$(1+\eps)$-approximate solution for any instance of weighted flow time in$O(n^{O(\ln W \ln P/\eps^3)})$time; here$P$is the ratio of more >>> TR06-074 | 24th April 2006 Shahar Dobzinski, Noam Nisan #### Approximations by Computationally-Efficient VCG-Based Mechanisms We consider computationally-efficient incentive-compatible mechanisms that use the VCG payment scheme, and study how well they can approximate the social welfare in auction settings. We obtain a$2$-approximation for multi-unit auctions, and show that this is best possible, even though from a purely computational perspective an FPTAS exists. For combinatorial ... more >>> TR99-011 | 14th April 1999 Matthias Krause, Petr Savicky, Ingo Wegener #### Approximations by OBDDs and the variable ordering problem Ordered binary decision diagrams (OBDDs) and their variants are motivated by the need to represent Boolean functions in applications. Research concerning these applications leads also to problems and results interesting from theoretical point of view. In this paper, methods from communication complexity and information theory ... more >>> TR09-114 | 13th November 2009 Emanuele Viola #### Are all distributions easy? Complexity theory typically studies the complexity of computing a function$h(x) : \{0,1\}^n \to \{0,1\}^m$of a given input$x$. We advocate the study of the complexity of generating the distribution$h(x)$for uniform$x$, given random bits. Our main results are: \begin{itemize} \item There are explicit$AC^0$circuits of ... more >>> TR09-089 | 26th September 2009 Guy Rothblum, Salil Vadhan #### Are PCPs Inherent in Efficient Arguments? Starting with Kilian (STOC 92), several works have shown how to use probabilistically checkable proofs (PCPs) and cryptographic primitives such as collision-resistant hashing to construct very efficient argument systems (a.k.a. computationally sound proofs), for example with polylogarithmic communication complexity. Ishai et al. (CCC 07) raised the question of whether PCPs ... more >>> TR09-057 | 23rd June 2009 Yonatan Bilu, Nathan Linial #### Are stable instances easy? We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP--hard problems are easier to solve. In particular, whether there exist algorithms that solve correctly ... more >>> TR13-028 | 14th February 2013 Mrinal Kumar, Gaurav Maheshwari, Jayalal Sarma #### Arithmetic Circuit Lower Bounds via MaxRank We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results :$\bullet$As ... more >>> TR09-026 | 17th February 2009 Vikraman Arvind, Pushkar Joglekar #### Arithmetic Circuit Size, Identity Testing, and Finite Automata Let$\F\{x_1,x_2,\cdots,x_n\}$be the noncommutative polynomial ring over a field$\F$, where the$x_i$'s are free noncommuting formal variables. Given a finite automaton$\A$with the$x_i$'s as alphabet, we can define polynomials$\f( mod A)$and$\f(div A)$obtained by natural operations that we call ... more >>> TR08-048 | 8th April 2008 Meena Mahajan, B. V. Raghavendra Rao #### Arithmetic circuits, syntactic multilinearity, and the limitations of skew formulae Functions in arithmetic NC1 are known to have equivalent constant width polynomial degree circuits, but the converse containment is unknown. In a partial answer to this question, we show that syntactic multilinear circuits of constant width and polynomial degree can be depth-reduced, though the resulting circuits need not be ... more >>> TR08-062 | 11th June 2008 Manindra Agrawal, V. Vinay #### Arithmetic Circuits: A Chasm at Depth Four We show that proving exponential lower bounds on depth four arithmetic circuits imply exponential lower bounds for unrestricted depth arithmetic circuits. In other words, for exponential sized circuits additional depth beyond four does not help. We then show that a complete black-box derandomization of Identity Testing problem for depth four ... more >>> TR13-026 | 11th February 2013 Ankit Gupta, Pritish Kamath, Neeraj Kayal, Ramprasad Saptharishi #### Arithmetic circuits: A chasm at depth three Revisions: 1 We show that, over$\mathbb{C}$, if an$n$-variate polynomial of degree$d = n^{O(1)}$is computable by an arithmetic circuit of size$s$(respectively by an algebraic branching program of size$s$) then it can also be computed by a depth three circuit (i.e. a$\Sigma \Pi \Sigma$-circuit) of size ... more >>> TR99-008 | 19th March 1999 Eric Allender, Vikraman Arvind, Meena Mahajan #### Arithmetic Complexity, Kleene Closure, and Formal Power Series Revisions: 1 , Comments: 1 The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC^1 and GapL. More precisely, we apply the Kleene closure of languages and the formal power series operations of inversion and root ... more >>> TR01-095 | 12th November 2001 Hubie Chen #### Arithmetic Versions of Constant Depth Circuit Complexity Classes The boolean circuit complexity classes AC^0 \subseteq AC^0[m] \subseteq TC^0 \subseteq NC^1 have been studied intensely. Other than NC^1, they are defined by constant-depth circuits of polynomial size and unbounded fan-in over some set of allowed gates. One reason for interest in these classes is that they contain the ... more >>> TR07-087 | 11th July 2007 Nutan Limaye, Meena Mahajan, B. V. Raghavendra Rao #### Arithmetizing classes around NC^1 and L The parallel complexity class NC^1 has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. \cite{CMTV} considered arithmetizations of two of these classes, #NC^1 and #BWBP. We further this study to include arithmetization of other classes. In particular, we show that counting paths ... more >>> TR09-055 | 10th June 2009 Venkatesan Chakaravarthy, Sambuddha Roy #### Arthur and Merlin as Oracles We study some problems solvable in deterministic polynomial time given oracle access to the (promise version of) the Arthur-Merlin class. Our main results are the following: (i)$BPP^{NP}_{||} \subseteq P^{prAM}_{||}$; (ii)$S_2^p \subseteq P^{prAM}$. In addition to providing new upperbounds for the classes$S_2^p$and$BPP^{NP}_{||}$, these results are interesting ... more >>> TR97-054 | 17th November 1997 Ran Raz, Gábor Tardos, Oleg Verbitsky, Nikolay Vereshchagin #### Arthur-Merlin Games in Boolean Decision Trees It is well known that probabilistic boolean decision trees cannot be much more powerful than deterministic ones (N.~Nisan, SIAM Journal on Computing, 20(6):999--1007, 1991). Motivated by a question if randomization can significantly speed up a nondeterministic computation via a boolean decision tree, we address structural properties of Arthur-Merlin games in ... more >>> TR13-020 | 2nd February 2013 Tom Gur, Ran Raz #### Arthur-Merlin Streaming Complexity We study the power of Arthur-Merlin probabilistic proof systems in the data stream model. We show a canonical$\mathcal{AM}$streaming algorithm for a wide class of data stream problems. The algorithm offers a tradeoff between the length of the proof and the space complexity that is needed to verify it. ... more >>> TR09-001 | 26th November 2008 Venkatesan Guruswami #### Artin automorphisms, Cyclotomic function fields, and Folded list-decodable codes Algebraic codes that achieve list decoding capacity were recently constructed by a careful folding'' of the Reed-Solomon code. The `low-degree'' nature of this folding operation was crucial to the list decoding algorithm. We show how such folding schemes conducive to list decoding arise out of the Artin-Frobenius automorphism at primes ... more >>> TR03-038 | 15th May 2003 Julia Chuzhoy, Sudipto Guha, Sanjeev Khanna, Seffi Naor #### Asymmetric k-center is log^*n-hard to Approximate We show that the asymmetric$k$-center problem is$\Omega(\log^* n)$-hard to approximate unless${\rm NP} \subseteq {\rm DTIME}(n^{poly(\log \log n)})$. Since an$O(\log^* n)$-approximation algorithm is known for this problem, this essentially resolves the approximability of this problem. This is the first natural problem whose approximability threshold does not polynomially ... more >>> TR99-048 | 7th December 1999 Beate Bollig, Ingo Wegener #### Asymptotically Optimal Bounds for OBDDs and the Solution of Some Basic OBDD Problems Ordered binary decision diagrams (OBDDs) are nowadays the most common dynamic data structure or representation type for Boolean functions. Among the many areas of application are verification, model checking, and computer aided design. For many functions it is easy to estimate the OBDD ... more >>> TR95-026 | 7th June 1995 Claus-Peter Schnorr, Horst Helmut Hoerner #### Attacking the Chor-Rivest Cryptosystem by Improved Lattice Reduction We introduce new algorithms for lattice basis reduction that are improvements of the LLL-algorithm. We demonstrate the power of these algorithms by solving random subset sum problems of arbitrary density with 74 and 82 many weights, by breaking the Chor-Rivest cryptoscheme in dimensions 103 and 151 ... more >>> TR98-076 | 13th November 1998 Nader H. Bshouty, Jeffrey J. Jackson, Christino Tamon #### Attribute Efficient PAC Learning of DNF with Membership Queries under the Uniform Distribution We study attribute efficient learning in the PAC learning model with membership queries. First, we give an {\it attribute efficient} PAC-learning algorithm for DNF with membership queries under the uniform distribution. Previous algorithms for DNF have sample size polynomial in the number of attributes$n$. Our algorithm is the first ... more >>> TR12-056 | 1st May 2012 Rocco Servedio, Li-Yang Tan, Justin Thaler #### Attribute-Efficient Learning and Weight-Degree Tradeoffs for Polynomial Threshold Functions Revisions: 1 We study the challenging problem of learning decision lists attribute-efficiently, giving both positive and negative results. Our main positive result is a new tradeoff between the running time and mistake bound for learning length-$k$decision lists over$n$Boolean variables. When the allowed running time is relatively high, our new ... more >>> TR13-047 | 27th March 2013 Christian Glaßer, Dung Nguyen, Christian Reitwießner, Alan Selman, Maximilian Witek #### Autoreducibility of Complete Sets for Log-Space and Polynomial-Time Reductions Comments: 1 We investigate the autoreducibility and mitoticity of complete sets for several classes with respect to different polynomial-time and logarithmic-space reducibility notions. Previous work in this area focused on polynomial-time reducibility notions. Here we obtain new mitoticity and autoreducibility results for the classes EXP and NEXP with respect to some restricted ... more >>> TR05-011 | 21st December 2004 Christian Glaßer, Mitsunori Ogihara, A. Pavan, Alan L. Selman, Liyu Zhang #### Autoreducibility, Mitoticity, and Immunity We show the following results regarding complete sets: NP-complete sets and PSPACE-complete sets are many-one autoreducible. Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are many-one autoreducible. EXP-complete sets are many-one mitotic. NEXP-complete sets are weakly many-one mitotic. PSPACE-complete sets are weakly Turing-mitotic. If ... more >>> TR13-054 | 4th April 2013 Yuval Filmus, Toniann Pitassi, Robert Robere, Stephen A. Cook #### Average Case Lower Bounds for Monotone Switching Networks An approximate computation of a Boolean function by a circuit or switching network is a computation which computes the function correctly on the majority of the inputs (rather than on all inputs). Besides being interesting in their own right, lower bounds for approximate computation have proved useful in many subareas ... more >>> TR95-019 | 14th April 1995 Jin-Yi Cai, Alan L. Selman #### Average time complexity classes TR06-073 | 8th June 2006 Andrej Bogdanov, Luca Trevisan #### Average-Case Complexity Revisions: 1 We survey the theory of average-case complexity, with a focus on problems in NP. more >>> TR03-031 | 8th April 2003 Birgit Schelm #### Average-Case Complexity Theory of Approximation Problems Both average-case complexity and the study of the approximability properties of NP-optimization problems are well established and active fields of research. By applying the notion of average-case complexity to approximation problems we provide a formal framework that allows the classification of NP-optimization problems according to their average-case approximability. Thus, known ... more >>> TR98-037 | 29th June 1998 Johannes Köbler, Rainer Schuler #### Average-Case Intractability vs. Worst-Case Intractability We use the assumption that all sets in NP (or other levels of the polynomial-time hierarchy) have efficient average-case algorithms to derive collapse consequences for MA, AM, and various subclasses of P/poly. As a further consequence we show for C in {P(PP), PSPACE} that ... more >>> TR12-062 | 17th May 2012 Ilan Komargodski, Ran Raz #### Average-Case Lower Bounds for Formula Size Revisions: 2 We give an explicit function$h:\{0,1\}^n\to\{0,1\}$such that any deMorgan formula of size$O(n^{2.499})$agrees with$h$on at most$\frac{1}{2} + \epsilon$fraction of the inputs, where$\epsilon$is exponentially small (i.e.$\epsilon = 2^{-n^{\Omega(1)}}$). Previous lower bounds for formula size were obtained for exact computation. The same technique ... more >>> TR11-006 | 20th January 2011 Sebastian Müller, Iddo Tzameret #### Average-Case Separation in Proof Complexity: Short Propositional Refutations for Random 3CNF Formulas Revisions: 1 Separating different propositional proof systems---that is, demonstrating that one proof system cannot efficiently simulate another proof system---is one of the main goals of proof complexity. Nevertheless, all known separation results between non-abstract proof systems are for specific families of hard tautologies: for what we know, in the average case all ... more >>> TR10-055 | 31st March 2010 Eric Allender #### Avoiding Simplicity is Complex Revisions: 2 It is a trivial observation that every decidable set has strings of length$n$with Kolmogorov complexity$\log n + O(1)$if it has any strings of length$n\$ at all. Things become much more interesting when one asks whether a similar property holds when one
considers *resource-bounded* Kolmogorov complexity. ... more >>>

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