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.2018.10
URN: urn:nbn:de:0030-drops-83418
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8341/
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Beame, Paul ; Fleming, Noah ; Impagliazzo, Russell ; Kolokolova, Antonina ; Pankratov, Denis ; Pitassi, Toniann ; Robere, Robert

Stabbing Planes

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Abstract

We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by branching on an inequality and its "integer negation." That is, we can (nondeterministically choose) a hyperplane a x >= b with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying a x >= b, the points satisfying a x <= b-1, and the middle slab b-1 < a x < b. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show somewhat surprisingly that Stabbing Planes can efficiently simulate Cutting Planes, and moreover, is strictly stronger than Cutting Planes under a reasonable conjecture. We prove linear lower bounds on the rank of Stabbing Planes refutations, by adapting
a lifting argument in communication complexity.

BibTeX - Entry

@InProceedings{beame_et_al:LIPIcs:2018:8341,
  author =	{Paul Beame and Noah Fleming and Russell Impagliazzo and Antonina Kolokolova and Denis Pankratov and Toniann Pitassi and Robert Robere},
  title =	{{Stabbing Planes}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Anna R. Karlin},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8341},
  URN =		{urn:nbn:de:0030-drops-83418},
  doi =		{10.4230/LIPIcs.ITCS.2018.10},
  annote =	{Keywords: Complexity Theory, Proof Complexity, Communication Complexity, Cutting Planes, Semi-Algebraic Proof Systems, Pseudo Boolean Solvers, SAT solvers, Inte}
}

Keywords: Complexity Theory, Proof Complexity, Communication Complexity, Cutting Planes, Semi-Algebraic Proof Systems, Pseudo Boolean Solvers, SAT solvers, Inte
Collection: 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)
Issue Date: 2018
Date of publication: 12.01.2018

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