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

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Theory and Applications of Models of Computation (TAMC 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6108))

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.

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Author information

Authors and Affiliations

  1. Computer Sciences Department, University of Wisconsin at Madison, Madison, WI, 53706, USA

    Jin-Yi Cai

  2. Department of Mathematics and Computer Science, Northern Michigan University, Marquette, MI, 49855, USA

    Michael Kowalczyk

Authors
  1. Jin-Yi Cai
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  2. Michael Kowalczyk
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Editor information

Editors and Affiliations

  1. Charles University, Malostranske nam 25, 11800, Praha 1, Czech Republic

    Jan Kratochvíl , Jiří Fiala  & Petr Kolman ,  & 

  2. State Key Lab. of Computer Science, Institute of Software,, Chinese Academy of Sciences, 100190, Beijing, P.R. China

    Angsheng Li

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Cai, JY., Kowalczyk, M. (2010). A Dichotomy for k-Regular Graphs with {0, 1}-Vertex Assignments and Real Edge Functions. In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds) Theory and Applications of Models of Computation. TAMC 2010. Lecture Notes in Computer Science, vol 6108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13562-0_30

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  • DOI: https://doi.org/10.1007/978-3-642-13562-0_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13561-3

  • Online ISBN: 978-3-642-13562-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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