A dichotomy for k-regular graphs with {0, 1}-vertex assignments and real edge functions

  • Authors:

  • Jin-Yi Cai;Michael Kowalczyk

  • Affiliations:

  • Computer Sciences Department, University of Wisconsin at Madison, Madison, WI;Department of Mathematics and Computer Science, Northern Michigan University, Marquette, MI

  • Venue:
  • TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
  • Year:
  • 2010

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Abstract

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.