License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2019.8
URN: urn:nbn:de:0030-drops-101010
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/10101/
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Bajpai, Swapnam ; Krishan, Vaibhav ; Kush, Deepanshu ; Limaye, Nutan ; Srinivasan, Srikanth

A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates

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LIPIcs-ITCS-2019-8.pdf (0.6 MB)

Abstract

We show that there is a zero-error randomized algorithm that, when given a small constant-depth Boolean circuit C made up of gates that compute constant-degree Polynomial Threshold functions or PTFs (i.e., Boolean functions that compute signs of constant-degree polynomials), counts the number of satisfying assignments to C in significantly better than brute-force time.
Formally, for any constants d,k, there is an epsilon > 0 such that the zero-error randomized algorithm counts the number of satisfying assignments to a given depth-d circuit C made up of k-PTF gates such that C has size at most n^{1+epsilon}. The algorithm runs in time 2^{n-n^{Omega(epsilon)}}.
Before our result, no algorithm for beating brute-force search was known for counting the number of satisfying assignments even for a single degree-k PTF (which is a depth-1 circuit of linear size).
The main new tool is the use of a learning algorithm for learning degree-1 PTFs (or Linear Threshold Functions) using comparison queries due to Kane, Lovett, Moran and Zhang (FOCS 2017). We show that their ideas fit nicely into a memoization approach that yields the #SAT algorithms.

BibTeX - Entry

@InProceedings{bajpai_et_al:LIPIcs:2018:10101,
  author =	{Swapnam Bajpai and Vaibhav Krishan and Deepanshu Kush and Nutan Limaye and Srikanth Srinivasan},
  title =	{{A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates}},
  booktitle =	{10th Innovations in Theoretical Computer Science  Conference (ITCS 2019)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{124},
  editor =	{Avrim Blum},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/10101},
  URN =		{urn:nbn:de:0030-drops-101010},
  doi =		{10.4230/LIPIcs.ITCS.2019.8},
  annote =	{Keywords: SAT, Polynomial Threshold Functions, Constant-depth Boolean Circuits, Linear Decision Trees, Zero-error randomized algorithms}
}

Keywords: SAT, Polynomial Threshold Functions, Constant-depth Boolean Circuits, Linear Decision Trees, Zero-error randomized algorithms
Collection: 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)
Issue Date: 2018
Date of publication: 08.01.2019

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