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Holant Problems for Regular Graphs with Complex Edge Functions

Authors Michael Kowalczyk, Jin-Yi Cai



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Michael Kowalczyk
Jin-Yi Cai

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Michael Kowalczyk and Jin-Yi Cai. Holant Problems for Regular Graphs with Complex Edge Functions. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 525-536, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)
https://doi.org/10.4230/LIPIcs.STACS.2010.2482

Abstract

We prove a complexity dichotomy theorem for Holant Problems on $3$-regular graphs with an arbitrary complex-valued edge function. Three new techniques are introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue Shifted Pairs, which allow us to prove that a pair of combinatorial gadgets \emph{in combination} succeed in proving \#P-hardness; and (3) algebraic symmetrization, which significantly lowers the \emph{symbolic complexity} of the proof for computational complexity. With \emph{holographic reductions} the classification theorem also applies to problems beyond the basic model.
Keywords
  • Computational complexity

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