The Complexity of Planar Boolean #CSP with Complex Weights

  • Heng Guo
  • Tyson Williams
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)


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.


Planar Graph Dichotomy Theorem Signature Matrix Constraint Graph Complex Weight 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Heng Guo
    • 1
  • Tyson Williams
    • 1
  1. 1.University of Wisconsin-MadisonMadisonUSA

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