This paper continues the investigation of the connection between proof
systems and approximation. The emphasis is on proving ``tight''
non-approximability results via consideration of measures like the
``free bit complexity'' and the ``amortized free bit complexity'' of
proof systems.

The first part of the paper presents a collection of new ... more >>>

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 $<{1\over 2}$.

This paper strengthens the low-error PCP characterization of NP, coming
closer to the ultimate BGLR conjecture. Namely, we prove that witnesses for
membership in any NP language can be verified with a constant
number of accesses, and with an error probability exponentially
small in the number ... more >>>

We give a new proof showing that it is NP-hard to color a 3-colorable
graph using just four colors. This result is already known (Khanna,
Linial, Safra 1992), but our proof is novel as it does not rely on
the PCP theorem, while the earlier one does. This ... more >>>

We prove that Minimum vertex cover on 4-regular hyper-graphs (or
in other words, hitting set where all sets have size exactly 4),
is hard to approximate within 2 - \epsilon.
We also prove that the maximization version, in which we
are allowed to pick B = ... more >>>

Using known results regarding PCP,
we present simple proofs of the inapproximability
of vertex cover for hypergraphs.
Specifically, we show that

(1) Approximating the size of the minimum vertex cover
in $O(1)$-regular hypergraphs to within a factor of~1.99999 is NP-hard.
(2) Approximating the size of ... more >>>

Given a $k$-uniform hypergraph, the E$k$-Vertex-Cover problem is
to find a minimum subset of vertices that ``hits'' every edge. We
show that for every integer $k \geq 5$, E$k$-Vertex-Cover is
NP-hard to approximate within a factor of $(k-3-\epsilon)$, for
an arbitrarily small constant $\epsilon > 0$.

This almost matches the ... more >>>

Locally testable codes are error-correcting codes that admit
very efficient codeword tests. Specifically, using a constant
number of (random) queries, non-codewords are rejected with
probability proportional to their distance from the code.

Locally testable codes are believed to be the combinatorial
core of PCPs. However, the relation is less ... more >>>

We continue the investigation of locally testable codes, i.e.,
error-correcting codes for whom membership of a given word in the
code can be tested probabilistically by examining it in very few
locations. We give two general results on local testability:
First, motivated by the recently proposed notion of robust
probabilistically ... more >>>

We give constructions of PCPs of length n*polylog(n) (with respect
to circuits of size n) that can be verified by making polylog(n)
queries to bits of the proof. These PCPs are not only shorter than
previous ones, but also simpler. Our (only) building blocks are
Reed-Solomon codes and the bivariate ... more >>>

The notion of promise problems was introduced and initially studied
by Even, Selman and Yacobi
(Information and Control, Vol.~61, pages 159-173, 1984).
In this article we survey some of the applications that this
notion has found in the twenty years that elapsed.
These include the notion of ... more >>>

Let C={c_1,...,c_n} be a set of constraints over a set of
variables. The {\em satisfiability-gap} of C is the smallest
fraction of unsatisfied constraints, ranging over all possible
assignments for the variables.

We prove a new combinatorial amplification lemma that doubles the
satisfiability-gap of a constraint-system, with only a linear ... more >>>

Given a function f:F^m \rightarrow F over a finite
field F, a low degree tester tests its proximity to
an m-variate polynomial of total degree at most d
over F. The tester is usually given access to an oracle
A providing the supposed restrictions of f to
affine subspaces of ... more >>>

We highlight a common theme in four relatively recent works
that establish remarkable results by an iterative approach.
Starting from a trivial construct,
each of these works applies an ingeniously designed
sequence of iterations that yields the desired result,
which is highly non-trivial. Furthermore, in each iteration,
the ... more >>>

We show a construction of a PCP with both sub-constant error and
almost-linear size. Specifically, for some constant alpha in (0,1),
we construct a PCP verifier for checking satisfiability of
Boolean formulas that on input of size n uses log n + O((log
n)^{1-alpha}) random bits to query a constant ... more >>>

An interactive-PCP (say, for the membership $x \in L$) is a
proof that can be verified by reading only one of its bits, with the
help of a very short interactive-proof.
We show that for membership in some languages $L$, there are
interactive-PCPs that are significantly shorter than the known
more >>>

In the undirected Edge-Disjoint Paths problem with Congestion
(EDPwC), we are given an undirected graph with $V$ nodes, a set of
terminal pairs and an integer $c$. The objective is to route as many
terminal pairs as possible, subject to the constraint that at most
$c$ demands can be routed ... more >>>

We study the approximability of the \maxcsp problem over non-boolean domains, more specifically over $\{0,1,\ldots,q-1\}$ for some integer $q$. We obtain a approximation algorithm that achieves a ratio of $C(q) \cdot k/q^k$ for some constant $C(q)$ depending only on $q$. Further, we extend the techniques of Samorodnitsky and Trevisan to ... more >>>

We define a non-uniform model of PCPs of Proximity, and observe that in this model the non-uniform verifiers can always be made very efficient. Specifically, we show that any non-uniform verifier can be modified to run in time that is roughly polynomial in its randomness and query complexity.

The main result of this paper is a simple, yet generic, composition theorem for low error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was well-understood only in the constant error regime. Existing composition methods in the low error regime were non-modular (i.e., very much tailored ... more >>>

The well known dichotomy conjecture of Feder and
Vardi states that for every &#64257;nite family &#915; of constraints CSP(&#915;) is
either polynomially solvable or NP-hard. Bulatov and Jeavons re-
formulated this conjecture in terms of the properties of the algebra
P ol(&#915;), where the latter is ... more >>>

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a $k$-colorable graph with $k$ colors so that a maximum fraction of edges are properly colored (i.e., their endpoints receive different colors). A random $k$-coloring properly colors an expected fraction ... more >>>

The Unique Games Conjecture (UGC) is possibly the most important open problem in the research of PCPs and hardness of approximation. The conjecture is a strengthening of the PCP Theorem, predicting the existence of a special type of PCP verifiers: 2-query verifiers that only make unique tests. Moreover, the UGC ... more >>>

We study Locally Testable Codes (LTCs) that can be tested by making two queries to the tested word using an affine test. That is, we consider LTCs over a finite field F, with codeword testers that only use tests of the form $av_i + bv_j = c$, where v is ... more >>>

A PCP is a proof system for NP in which the proof can be checked by a probabilistic verifier. The verifier is only allowed to read a very small portion of the proof, and in return is allowed to err with some bounded probability. The probability that the verifier accepts ... more >>>

In this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each
equation contains at most three variables.

Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be ``non-trivial". Here is
an informal statement of our ... more >>>

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 >>>

The PCP theorem asserts the existence of proofs that can be verified by a verifier that reads only a very small part of the proof. The theorem was originally proved by Arora and Safra (J. ACM 45(1)) and Arora et al. (J. ACM 45(3)) using sophisticated algebraic tools. More than ... more >>>

In this paper we put forward a conjecture: an instantiation of the Sliding Scale Conjecture of Bellare, Goldwasser, Lund and Russell to projection games. We refer to this conjecture as the Projection Games Conjecture.

We further suggest the research agenda of establishing new hardness of approximation results based on the ... more >>>