{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11540","name":"Satisfiability and Derandomization for Small Polynomial Threshold Circuits","abstract":"A polynomial threshold function (PTF) is defined as the sign of a polynomial p : {0,1}^n ->R. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth.\n - Satisfiability (#SAT). We give the first zero-error randomized algorithm faster than exhaustive search that counts the number of satisfying assignments of a given constant-depth circuit with a super-linear number of wires whose gates are s-sparse PTFs, for s almost quadratic in the input size of the circuit; here a PTF is called s-sparse if its underlying polynomial has at most s monomials. More specifically, we show that, for any large enough constant c, given a depth-d circuit with (n^{2-1\/c})-sparse PTF gates that has at most n^{1+epsilon_d} wires, where epsilon_d depends only on c and d, the number of satisfying assignments of the circuit can be computed in randomized time 2^{n-n^{epsilon_d}} with zero error. This generalizes the result by Chen, Santhanam and Srinivasan (CCC, 2016) who gave a SAT algorithm for constant-depth circuits of super-linear wire complexity with linear threshold function (LTF) gates only.\n - Quantified derandomization. The quantified derandomization problem, introduced by Goldreich and Wigderson (STOC, 2014), asks to compute the majority value of a given Boolean circuit, under the promise that the minority-value inputs to the circuit are very few. We give a quantified derandomization algorithm for constant-depth PTF circuits with a super-linear number of wires that runs in quasi-polynomial time. More specifically, we show that for any sufficiently large constant c, there is an algorithm that, given a degree-Delta PTF circuit C of depth d with n^{1+1\/c^d} wires such that C has at most 2^{n^{1-1\/c}} minority-value inputs, runs in quasi-polynomial time exp ((log n)^{O (Delta^2)}) and determines the majority value of C. (We obtain a similar quantified derandomization result for PTF circuits with n^{Delta}-sparse PTF gates.) This extends the recent result of Tell (STOC, 2018) for constant-depth LTF circuits of super-linear wire complexity.\n - Pseudorandom generators. We show how the classical Nisan-Wigderson (NW) generator (JCSS, 1994) yields a nontrivial pseudorandom generator for PTF circuits (of unrestricted depth) with sub-linearly many gates. As a corollary, we get a PRG for degree-Delta PTFs with the seed length exp (sqrt{Delta * log n})* log^2(1\/epsilon).","keywords":["constant-depth circuits","polynomial threshold functions","circuit analysis algorithms","SAT","derandomization","quantified derandomization","pseudorandom generators."],"author":[{"@type":"Person","name":"Kabanets, Valentine","givenName":"Valentine","familyName":"Kabanets","affiliation":"School of Computing Science, Simon Fraser University, Burnaby, BC, Canada"},{"@type":"Person","name":"Lu, Zhenjian","givenName":"Zhenjian","familyName":"Lu","affiliation":"School of Computing Science, Simon Fraser University, Burnaby, BC, Canada"}],"position":46,"pageStart":"46:1","pageEnd":"46:19","dateCreated":"2018-08-13","datePublished":"2018-08-13","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kabanets, Valentine","givenName":"Valentine","familyName":"Kabanets","affiliation":"School of Computing Science, Simon Fraser University, Burnaby, BC, Canada"},{"@type":"Person","name":"Lu, Zhenjian","givenName":"Zhenjian","familyName":"Lu","affiliation":"School of Computing Science, Simon Fraser University, Burnaby, BC, Canada"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2018.46","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/dx.doi.org\/10.1137\/1.9780898718539","isPartOf":{"@type":"PublicationVolume","@id":"#volume6319","volumeNumber":116,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2018)","dateCreated":"2018-08-13","datePublished":"2018-08-13","editor":[{"@type":"Person","name":"Blais, Eric","givenName":"Eric","familyName":"Blais"},{"@type":"Person","name":"Jansen, Klaus","givenName":"Klaus","familyName":"Jansen"},{"@type":"Person","name":"D. P. Rolim, Jos\u00e9","givenName":"Jos\u00e9","familyName":"D. P. Rolim"},{"@type":"Person","name":"Steurer, David","givenName":"David","familyName":"Steurer"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article11540","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6319"}}}