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DOI: 10.4230/LIPIcs.STACS.2011.249

The Complexity of Weighted Boolean #CSP Modulo k

Authors: Heng Guo, Sangxia Huang, Pinyan Lu, and Mingji Xia

Published in: LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

Abstract

We prove a complexity dichotomy theorem for counting weighted Boolean CSP modulo k for any positive integer $k>1$. This generalizes a theorem by Faben for the unweighted setting. In the weighted setting, there are new interesting tractable problems. We first prove a dichotomy theorem for the finite field case where k is a prime. It turns out that the dichotomy theorem for the finite field is very similar to the one for the complex weighted Boolean #CSP, found by [Cai, Lu and Xia, STOC 2009]. Then we further extend the result to an arbitrary integer k.

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Heng Guo, Sangxia Huang, Pinyan Lu, and Mingji Xia. The Complexity of Weighted Boolean #CSP Modulo k. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 249-260, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{guo_et_al:LIPIcs.STACS.2011.249,
  author =	{Guo, Heng and Huang, Sangxia and Lu, Pinyan and Xia, Mingji},
  title =	{{The Complexity of Weighted Boolean #CSP Modulo k}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)},
  pages =	{249--260},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Schwentick, Thomas and D\"{u}rr, Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.249},
  URN =		{urn:nbn:de:0030-drops-30158},
  doi =		{10.4230/LIPIcs.STACS.2011.249},
  annote =	{Keywords: #CSP, dichotomy theorem, counting problems, computational complexity}
}

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Extended Abstract

DOI: 10.4230/LIPIcs.STACS.2008.1368

A Theory for Valiant's Matchcircuits (Extended Abstract)

Authors: Angsheng Li and Mingji Xia

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

Abstract

The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every $k$, the nonsingular character matrices of $k$-bit matchgates form a group, extending the recent work of Cai and Choudhary (2006) of the same result for the case of $k=2$, and that the single and the two-bit matchgates are universal for matchcircuits, answering a question of Valiant (2002).

Cite as

Angsheng Li and Mingji Xia. A Theory for Valiant's Matchcircuits (Extended Abstract). In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 491-502, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{li_et_al:LIPIcs.STACS.2008.1368,
  author =	{Li, Angsheng and Xia, Mingji},
  title =	{{A Theory for Valiant's Matchcircuits}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{491--502},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1368},
  URN =		{urn:nbn:de:0030-drops-13686},
  doi =		{10.4230/LIPIcs.STACS.2008.1368},
  annote =	{Keywords: Pfaffian, Matchgate, Matchcircuit}
}

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Documents authored by Xia, Mingji

The Complexity of Weighted Boolean #CSP Modulo k

Abstract

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A Theory for Valiant's Matchcircuits (Extended Abstract)

Abstract

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