Holant Problems for Regular Graphs with Complex Edge Functions

Kowalczyk, Michael; Cai, Jin-Yi

doi:10.4230/LIPIcs.STACS.2010.2482

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Holant Problems for Regular Graphs with Complex Edge Functions

Authors Michael Kowalczyk, Jin-Yi Cai

Part of: Volume: 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010)
Part of: Series: Leibniz International Proceedings in Informatics (LIPIcs)
Part of: Conference: Symposium on Theoretical Aspects of Computer Science (STACS)
License: Creative Commons Attribution-NoDerivs 3.0 Unported license
Publication Date: 2010-03-09

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DOI: 10.4230/LIPIcs.STACS.2010.2482
URN: urn:nbn:de:0030-drops-24826

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Computational complexity

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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.

Cite As Get BibTex

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

Author Details

Michael Kowalczyk

Jin-Yi Cai

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