A Dichotomy for k-Regular Graphs with {0, 1}-Vertex Assignments and Real Edge Functions

  • Jin-Yi Cai
  • Michael Kowalczyk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6108)


We prove a complexity dichotomy theorem for a class of Holant Problems over k-regular graphs, for any fixed k. These problems can be viewed as graph homomorphisms from an arbitrary k-regular input graph G to the weighted two vertex graph on {0,1} defined by a symmetric function h. We completely classify the computational complexity of this problem. We show that there are exactly the following alternatives, for any given h. Depending on h, over k-regular graphs: Either (1) the problem is #P-hard even for planar graphs; or (2) the problem is #P-hard for general (non-planar) graphs, but solvable in polynomial time for planar graphs; or (3) the problem is solvable in polynomial time for general graphs. The dependence on h is an explicit criterion. Furthermore, we show that in case (2) the problem is solvable in polynomial time over k-regular planar graphs, by exactly the theory of holographic algorithms using matchgates.


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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Michael Kowalczyk
    • 2
  1. 1.Computer Sciences DepartmentUniversity of Wisconsin at MadisonMadisonUSA
  2. 2.Department of Mathematics and Computer ScienceNorthern Michigan UniversityMarquetteUSA

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