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DOI: 10.4230/LIPIcs.ITCS.2019.8
A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates

Authors: Swapnam Bajpai, Vaibhav Krishan, Deepanshu Kush, Nutan Limaye, and Srikanth Srinivasan

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

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.
Cite as

Swapnam Bajpai, Vaibhav Krishan, Deepanshu Kush, Nutan Limaye, and Srikanth Srinivasan. A #SAT Algorithm for Small Constant-Depth Circuits with PTF Gates. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bajpai_et_al:LIPIcs.ITCS.2019.8,
  author =	{Bajpai, Swapnam and Krishan, Vaibhav and Kush, Deepanshu and Limaye, Nutan and Srinivasan, Srikanth},
  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 =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.8},
  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}
}
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