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.ICALP.2016.47
URN: urn:nbn:de:0030-drops-63279
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6327/
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Curticapean, Radu

Parity Separation: A Scientifically Proven Method for Permanent Weight Loss

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LIPIcs-ICALP-2016-47.pdf (0.6 MB)

Abstract

Given an edge-weighted graph G, let PerfMatch(G) denote the weighted sum over all perfect matchings M in G, weighting each matching M by the product of weights of edges in M. If G is unweighted, this plainly counts the perfect matchings of G.

In this paper, we introduce parity separation, a new method for reducing PerfMatch to unweighted instances: For graphs G with edge-weights 1 and -1, we construct two unweighted graphs G1 and G2 such that PerfMatch(G) = PerfMatch(G1) - PerfMatch(G2). This yields a novel weight removal technique for counting perfect matchings, in addition to those known from classical #P-hardness proofs. Our technique is based upon the Holant framework and matchgates. We derive the following applications:

Firstly, an alternative #P-completeness proof for counting unweighted perfect matchings.

Secondly, C=P-completeness for deciding whether two given unweighted graphs have the same number of perfect matchings. To the best of our knowledge, this is the first C=P-completeness result for the “equality-testing version” of any natural counting problem that is not already #P-hard under parsimonious reductions.

Thirdly, an alternative tight lower bound for counting unweighted perfect matchings under the counting exponential-time hypothesis #ETH.

BibTeX - Entry

@InProceedings{curticapean:LIPIcs:2016:6327,
  author =	{Radu Curticapean},
  title =	{{Parity Separation: A Scientifically Proven Method for Permanent Weight Loss}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{47:1--47:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6327},
  URN =		{urn:nbn:de:0030-drops-63279},
  doi =		{10.4230/LIPIcs.ICALP.2016.47},
  annote =	{Keywords: perfect matchings, counting complexity, structural complexity, exponentialtime hypothesis}
}

Keywords: perfect matchings, counting complexity, structural complexity, exponentialtime hypothesis
Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 23.08.2016

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