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The Complexity of Planar Boolean #CSP with Complex Weights

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Automata, Languages, and Programming (ICALP 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7965))

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Abstract

We prove a complexity dichotomy theorem for symmetric complex-weighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #P-hard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also obtain a dichotomy theorem for a symmetric arity 4 signature with complex weights in the planar Holant framework, which we use in the proof of our #CSP dichotomy. In particular, we reduce the problem of evaluating the Tutte polynomial of a planar graph at the point (3,3) to counting the number of Eulerian orientations over planar 4-regular graphs to show the latter is #P-hard. This strengthens a theorem by Huang and Lu to the planar setting.

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Authors and Affiliations

  1. University of Wisconsin-Madison, Madison, WI, USA

    Heng Guo & Tyson Williams

Authors
  1. Heng Guo
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  2. Tyson Williams
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Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Postboks 7803, 5020, Bergen, Norway

    Fedor V. Fomin

  2. Faculty of Computing, University of Latvia, Raina bulv. 19, 1586, Riga, Latvia

    Rūsiņš Freivalds

  3. Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Marta Kwiatkowska

  4. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, 76100, Rehovot, Israel

    David Peleg

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Guo, H., Williams, T. (2013). The Complexity of Planar Boolean #CSP with Complex Weights. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39206-1_44

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  • DOI: https://doi.org/10.1007/978-3-642-39206-1_44

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39205-4

  • Online ISBN: 978-3-642-39206-1

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