Results 11  20
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37
Local reductions
, 2013
"... We reduce nondeterministic time T ≥ 2 n to a 3SAT instance φ of size φ  = T ·log O(1) T such that there is an explicit circuit C that on input an index i of logφ bits outputs the ith clause, and each output bit of C depends on O(1) inputs bits. The previous best result was C in NC 1. Even in th ..."
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We reduce nondeterministic time T ≥ 2 n to a 3SAT instance φ of size φ  = T ·log O(1) T such that there is an explicit circuit C that on input an index i of logφ bits outputs the ith clause, and each output bit of C depends on O(1) inputs bits. The previous best result was C in NC 1. Even in the simpler setting of φ  = poly(T) the previous best result was C in AC 0. More generally, for any time T ≥ n and parameter r ≤ n we obtain log 2φ  = max(logT,n/r)+O(logn)+O(loglogT) and each output bit of C is a decision tree of depth O(logr). As an application, we simplify the proof of Williams ’ ACC 0 lower bound, and tighten his connection between satisfiability algorithms and lower bounds.
On MediumUniformity and Circuit Lower Bounds
"... Abstract—We explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts: 1. Lower Bounds Against MediumUniform Circuits. Informally, a circuit class is “medium uniform ” if it can be genera ..."
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Abstract—We explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts: 1. Lower Bounds Against MediumUniform Circuits. Informally, a circuit class is “medium uniform ” if it can be generated by an algorithmic process that is somewhat complex (stronger than LOGTIME) but not infeasible. Using a new kind of indirect diagonalization argument, we prove several new unconditional lower bounds against medium uniform circuit classes, including: • For all k, P is not contained in Puniform SIZE(n k). That is, for all k there is a language Lk ∈ P that does not have O(n k)size circuits constructible in polynomial time. This improves Kannan’s lower bound from 1982 that NP is not in Puniform SIZE(n k) for any fixed k.
Marginal Hitting Sets Imply SuperPolynomial Lower Bounds for Permanent
"... Suppose f is a univariate polynomial of degree r = r(n) that is computed by a size n arithmetic circuit. It is a basic fact of algebra that a nonzero univariate polynomial of degree r can vanish on at most r points. This implies that for checking whether f is identically zero, it suffices to query f ..."
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Suppose f is a univariate polynomial of degree r = r(n) that is computed by a size n arithmetic circuit. It is a basic fact of algebra that a nonzero univariate polynomial of degree r can vanish on at most r points. This implies that for checking whether f is identically zero, it suffices to query f on an arbitrary test set of r + 1 points. Could this bruteforce method be improved upon by a single point? We develop a framework where such a marginal improvement implies that Permanent does not have polynomial size arithmetic circuits. More formally, we formulate the following hypothesis for any field of characteristic zero: There is a fixed depth d and some function s(n) = O(n), such that for arbitrarily small ɛ> 0, there exists a hitting set Hn ⊂ Z of size at most 2 s(nɛ) against univariate polynomials of degree at most 2 s(nɛ) computable by size n constantfree 1 arithmetic circuits, where Hn can be encoded by uniform TC 0 circuits of size 2 O(nɛ) and depth d. We prove that the hypothesis implies that Permanent does not have polynomial size constantfree arithmetic circuits. Our hypothesis provides a unifying perspective on several important complexity theoretic conjectures, as it follows from these conjectures for different degree ranges as determined by the function s(n). We will show that it follows for s(n) = n from the widelybelieved assumption that poly size Boolean circuits cannot compute the Permanent of a 0, 1matrix over Z. The hypothesis can also be easily derived from the ShubSmale τconjecture [21], for any s(n)
On the Limits of Sparsification
, 2012
"... Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for kCNFs: every kCNF is a subexponential size disjunction of kCNFs with a linear number of clauses. This lemma has subsequently played a key role in the study of the exact complexity of the satisfiability problem. A natural ..."
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Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for kCNFs: every kCNF is a subexponential size disjunction of kCNFs with a linear number of clauses. This lemma has subsequently played a key role in the study of the exact complexity of the satisfiability problem. A natural question is whether an analogous structural result holds for CNFs or even for broader nonuniform classes such as constantdepth circuits or Boolean formulae. We prove a very strong negative result in this connection: For every superlinear function f(n), there are CNFs of size f(n) which cannot be written as a disjunction of 2 n−εn CNFs each having a linear number of clauses for any ε> 0. We also give a hierarchy of such nonsparsifiable CNFs: For every k, there is a k ′ for which there are CNFs of size n k′ which cannot be written as a subexponential size disjunction of CNFs of size n k. Furthermore, our lower bounds hold not just against CNFs but against an arbitrary family
On beating the hybrid argument
"... The hybrid argument allows one to relate the distinguishability of a distribution (from uniform) to the predictability of individual bits given a prefix. The argument incurs a loss of a factor k equal to the bitlength of the distributions: ɛdistinguishability implies only ɛ/kpredictability. This ..."
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The hybrid argument allows one to relate the distinguishability of a distribution (from uniform) to the predictability of individual bits given a prefix. The argument incurs a loss of a factor k equal to the bitlength of the distributions: ɛdistinguishability implies only ɛ/kpredictability. This paper studies the consequences of avoiding this loss – what we call “beating the hybrid argument” – and develops new proof techniques that circumvent the loss in certain natural settings. Specifically, we obtain the following results: 1. We give an instantiation of the NisanWigderson generator (JCSS ’94) that can be broken by quantum computers, and that is o(1)unpredictable against AC 0. This is not enough to imply indistinguishability via the hybrid argument because of the hybridargument loss; nevertheless, we conjecture that this generator indeed fools AC 0, and we prove this statement for a simplified version of the problem. Our conjecture implies the existence of an oracle relative to which BQP is not in the PH, a longstanding open problem. 2. We show that the “INW” generator by Impagliazzo, Nisan, and Wigderson (STOC ’94) with seed length O(log n log log n) produces a distribution that is 1 / log nunpredictable
Short PCPs with projection queries
, 2014
"... We construct a PCP for NTIME(2 n) with constant soundness, 2 n poly(n) proof length, and poly(n) queries where the verifier’s computation is simple: the queries are a projection of the input randomness, and the computation on the prover’s answers is a 3CNF. The previous upper bound for these two com ..."
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We construct a PCP for NTIME(2 n) with constant soundness, 2 n poly(n) proof length, and poly(n) queries where the verifier’s computation is simple: the queries are a projection of the input randomness, and the computation on the prover’s answers is a 3CNF. The previous upper bound for these two computations was polynomialsize circuits. Composing this verifier with a proof oracle increases the circuitdepth of the latter by 2. Our PCP is a simple variant of the PCP by BenSasson, Goldreich, Harsha, Sudan, and Vadhan (CCC 2005). We also give a more modular exposition of the latter, separating the combinatorial from the algebraic arguments. If our PCP is taken as a black box, we obtain a more direct proof of the result by Williams, later with Santhanam (CCC 2013) that derandomizing circuits on n bits from a class C in time 2 n /n ω(1) yields that NEXP is not in a related circuit class C ′. Our proof yields a tighter connection: C is an AndOr of circuits from C ′. Along the way we show that the same lower bound follows if the satisfiability of the And of any 3 circuits from C ′ can be solved in time 2 n /n ω(1). ∗The research leading to these results has received funding from the European Community’s
Quantum walk speedup of backtracking algorithms
, 2015
"... We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attemp ..."
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We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on n variables, or outputs that none exists, and whose corresponding tree contains T vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a boundederror quantum algorithm which completes the same task using O( Tn3/2 log n) tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree. We also discuss how, for certain distributions on the inputs, the algorithm can lead to an averagecase exponential speedup. 1
Compression of Boolean Functions
, 2013
"... We consider the problem of compression for “easy ” Boolean functions: given the truth table of an nvariate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2n) a circuit C (no restriction on the type of C) computing f so tha ..."
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We consider the problem of compression for “easy ” Boolean functions: given the truth table of an nvariate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2n) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size 2n/n. We get both positive and negative results. On the positive side, we show that several circuit classes for which lower bounds are proved by a method of random restrictions: • AC0, • (de Morgan) formulas, and • (readonce) branching programs, allow nontrivial compression for circuits up to the size for which lower bounds are known. On the negative side, we show that compressing functions from any class C ⊆ P/poly implies superpolynomial lower bounds against C for a function in NEXP; we also observe that compressing monotone functions of polynomial circuit complexity or functions computable by largesize AC0 circuits would also imply new superpolynomial circuit lower bounds. Finally, we apply the ideas used for compression to get zeroerror randomized #SATalgorithms for de Morgan and completebasis formulas, as well as branching programs, on n variables of about quadratic size that run in expected time 2n/2n ϵ, for some ϵ> 0 (dependent on the size of the formula/branching program). ∗Research partially supported by an NSERC Discovery grant. †Research partially supported by an NSERC Discovery grant. 1