Holographic Algorithms Beyond Matchgates

  • Jin-Yi Cai
  • Heng Guo
  • Tyson Williams
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)


Holographic algorithms based on matchgates were introduced by Valiant. These algorithms run in polynomial-time and are intrinsically for planar problems. We introduce two new families of holographic algorithms, which work over general, i.e., not necessarily planar, graphs. The two underlying families of constraint functions are of the affine and product types. These play the role of Kasteleyn’s algorithm for counting planar perfect matchings. The new algorithms are obtained by transforming a problem to one of these two families by holographic reductions. We present a polynomial-time algorithm to decide if a given counting problem has a holographic algorithm using these constraint families. When the constraints are symmetric, we give a polynomial-time decision procedure in the size of the succinct presentation of symmetric constraint functions. This procedure shows that the recent dichotomy theorem for Holant problems with symmetric constraints is polynomial-time decidable.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Heng Guo
    • 1
  • Tyson Williams
    • 1
  1. 1.University of Wisconsin–MadisonMadisonUSA

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