Mihir Bellare, Oded Goldreich, Erez Petrank

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 ...
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Vikraman Arvind, Venkatesh Raman

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 ...
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Luca Trevisan

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 ...
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Ronen Shaltiel, Chris Umans

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 >>>

Luca Trevisan, Salil Vadhan, David Zuckerman

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 ... more >>>

Oded Goldreich

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 ...
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Prahladh Harsha, Adam Klivans, Raghu Meka

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 ...
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Dan Gutfreund, Akinori Kawachi

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 >>>

Parikshit Gopalan, Adam Klivans, Raghu Meka

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 >>>

Emanuele Viola

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 >>>

Albert Atserias, Moritz Müller, Sergi Oliva

The relativized weak pigeonhole principle states that if at least $2n$ out of $n^2$ pigeons fly into $n$ holes, then some hole must be doubly occupied. We prove that every DNF-refutation of the CNF encoding of this principle requires size $2^{(\log n)^{3/2-\epsilon}}$ for every $\epsilon > 0$ and every sufficiently ... more >>>

Oded Goldreich, Avi Wigderson

{\em Does derandomization of probabilistic algorithms become easier when the number of ``bad'' random inputs is extremely small?}

In relation to the above question, we put forward the following {\em quantified derandomization challenge}:

For a class of circuits $\cal C$ (e.g., P/poly or $AC^0,AC^0[2]$) and a bounding function $B:\N\to\N$ (e.g., ...
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Andrzej Dudek , Marek Karpinski, Andrzej Rucinski, Edyta Szymanska

We design a fully polynomial time approximation scheme (FPTAS) for counting the number of matchings (packings) in arbitrary 3-uniform hypergraphs of maximum degree three, referred to as $(3,3)$-hypergraphs. It is the first polynomial time approximation scheme for that problem, which includes also, as a special case, the 3D Matching counting ... more >>>

Moritz Müller, Ján Pich

We ask for feasibly constructive proofs of known circuit lower bounds for explicit functions on bit strings of length $n$. In 1995 Razborov showed that many can be proved in Cook’s theory $PV_1$, a bounded arithmetic formalizing polynomial time reasoning. He formalized circuit lower bound statements for small $n$ of ... more >>>

Scott Aaronson

We consider the following problem: estimate the size of a nonempty set $S\subseteq\left[ N\right] $, given both quantum queries to a membership oracle for $S$, and a device that generates equal superpositions $\left\vert S\right\rangle $ over $S$ elements. We show that, if $\left\vert S\right\vert $ is neither too large nor ... more >>>

William Kretschmer

We prove a query complexity lower bound for $QMA$ protocols that solve approximate counting: estimating the size of a set given a membership oracle. This gives rise to an oracle $A$ such that $SBP^A \not\subset QMA^A$, resolving an open problem of Aaronson [2]. Our proof uses the polynomial method to ... more >>>

Scott Aaronson, Robin Kothari, William Kretschmer, Justin Thaler

This paper proves new limitations on the power of quantum computers to solve approximate counting---that is, multiplicatively estimating the size of a nonempty set $S\subseteq [N]$.

Given only a membership oracle for $S$, it is well known that approximate counting takes $\Theta(\sqrt{N/|S|})$ quantum queries. But what if a quantum algorithm ... more >>>

Joshua Cook

We give two results on the size of AC0 circuits computing promise majority. $\epsilon$-promise majority is majority promised that either at most an $\epsilon$ fraction of the input bits are 1, or at most $\epsilon$ are 0.

First, we show super quadratic lower bounds on both monotone and general depth ... more >>>

Oded Goldreich

For any fixed $t$, we present two fine-grained reductions of the problem of approximately counting the number of $t$-cliques in a graph to the problem of detecting a $t$-clique in a graph.

One of our reductions is slightly better than the prior reduction of Dell, Lapinskas, and Meeks (SODA20) and ...
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