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.IPEC.2017.11
URN: urn:nbn:de:0030-drops-85459
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Cardinal, Jean ; Nummenpalo, Jerri ; Welzl, Emo

Solving and Sampling with Many Solutions: Satisfiability and Other Hard Problems

LIPIcs-IPEC-2017-11.pdf (0.5 MB)


We investigate parameterizing hard combinatorial problems by the size of the solution set compared to all solution candidates. Our main result is a uniform sampling algorithm for satisfying assignments of 2-CNF formulas that runs in expected time O^*(eps^{-0.617}) where eps is the fraction of assignments that are satisfying. This improves significantly over the trivial sampling bound of expected Theta^*(eps^{-1}), and on all previous algorithms whenever eps = Omega(0.708^n). We also consider algorithms for 3-SAT with an eps fraction of satisfying assignments, and prove that it can be solved in O^*(eps^{-2.27}) deterministic time, and in O^*(eps^{-0.936}) randomized time. Finally, to further demonstrate the applicability of this framework, we also explore how similar techniques can be used for vertex cover problems.

BibTeX - Entry

  author =	{Jean Cardinal and Jerri Nummenpalo and Emo Welzl},
  title =	{{Solving and Sampling with Many Solutions: Satisfiability and Other Hard Problems}},
  booktitle =	{12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
  pages =	{11:1--11:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-051-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{89},
  editor =	{Daniel Lokshtanov and Naomi Nishimura},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-85459},
  doi =		{10.4230/LIPIcs.IPEC.2017.11},
  annote =	{Keywords: Satisfiability, Sampling, Parameterized complexity}

Keywords: Satisfiability, Sampling, Parameterized complexity
Collection: 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)
Issue Date: 2018
Date of publication: 02.03.2018

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