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Electronic Colloquium on Computational Complexity
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REPORTS > KEYWORD > APPROXIMATE COUNTING:
Reports tagged with Approximate Counting:
TR98-032 | 10th June 1998
Mihir Bellare, Oded Goldreich, Erez Petrank

Uniform Generation of NP-witnesses using an NP-oracle.

A Uniform Generation procedure for $NP$ is an
algorithm which given any input in a fixed NP-language, outputs a uniformly
distributed NP-witness for membership of the input in the language.
We present a Uniform Generation procedure for $NP$ that runs in probabilistic
polynomial-time with an NP-oracle. This improves upon ... 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 >>>


TR02-069 | 14th November 2002
Luca Trevisan

A Note on Deterministic Approximate Counting for k-DNF

Revisions: 1

We describe a deterministic algorithm that, for constant k,
given a k-DNF or k-CNF formula f and a parameter e, runs in time
linear in the size of f and polynomial in 1/e and returns an
estimate of the fraction of satisfying assignments for f up to ... more >>>


TR04-086 | 12th October 2004
Ronen Shaltiel, Chris Umans

Pseudorandomness for Approximate Counting and Sampling

We study computational procedures that use both randomness and nondeterminism. Examples are Arthur-Merlin games and approximate counting and sampling of NP-witnesses. The goal of this paper is to derandomize such procedures under the weakest possible assumptions.

Our main technical contribution allows one to ``boost'' a given hardness assumption. One special ... more >>>


TR05-012 | 17th January 2005
Luca Trevisan, Salil Vadhan, David Zuckerman

Compression of Samplable Sources

We study the compression of polynomially samplable sources. In particular, we give efficient prefix-free compression and decompression algorithms for three classes of such sources (whose support is a subset of {0,1}^n).

1. We show how to compress sources X samplable by logspace machines to expected length H(X)+O(1).

Our next results ... more >>>


TR05-018 | 6th February 2005
Oded Goldreich

On Promise Problems (a survey in memory of Shimon Even [1935-2004])

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 >>>


TR09-144 | 24th December 2009
Prahladh Harsha, Adam Klivans, Raghu Meka

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 >>>


TR09-146 | 29th December 2009
Dan Gutfreund, Akinori Kawachi

Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

We show that if Arthur-Merlin protocols can be derandomized, then there is a Boolean function computable in deterministic exponential-time with access to an NP oracle, that cannot be computed by Boolean circuits of exponential size. More formally, if $\mathrm{prAM}\subseteq \mathrm{P}^{\mathrm{NP}}$ then there is a Boolean function in $\mathrm{E}^{\mathrm{NP}}$ that requires ... more >>>


TR10-133 | 20th August 2010
Parikshit Gopalan, Adam Klivans, Raghu Meka

Polynomial-Time Approximation Schemes for Knapsack and Related Counting Problems using Branching Programs

We give a deterministic, polynomial-time algorithm for approximately counting the number of {0,1}-solutions to any instance of the knapsack problem. On an instance of length n with total weight W and accuracy parameter eps, our algorithm produces a (1 + eps)-multiplicative approximation in time poly(n,log W,1/eps). We also give algorithms ... more >>>


TR10-175 | 14th November 2010
Emanuele Viola

Randomness buys depth for approximate counting

We show that the promise problem of distinguishing $n$-bit strings of hamming weight $\ge 1/2 + \Omega(1/\log^{d-1} n)$ from strings of weight $\le 1/2 - \Omega(1/\log^{d-1} n)$ can be solved by explicit, randomized (unbounded-fan-in) poly(n)-size depth-$d$ circuits with error $\le 1/3$, but cannot be solved by deterministic poly(n)-size depth-$(d+1)$ circuits, ... more >>>




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