when quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2010.2482
URN: urn:nbn:de:0030-drops-24826
URL: http://drops.dagstuhl.de/opus/volltexte/2010/2482/

Kowalczyk, Michael ; Cai, Jin-Yi

### Holant Problems for Regular Graphs with Complex Edge Functions

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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.

### BibTeX - Entry

@InProceedings{kowalczyk_et_al:LIPIcs:2010:2482,
author =	{Michael Kowalczyk and Jin-Yi Cai},
title =	{Holant Problems for Regular Graphs with Complex Edge Functions},
booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010)},
pages =	{525--536},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-16-3},
ISSN =	{1868-8969},
year =	{2010},
volume =	{5},
editor =	{Jean-Yves Marion and Thomas Schwentick},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2482},
doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2010.2482},
annote =	{Keywords: Computational complexity}
}


 Keywords: Computational complexity Seminar: 27th International Symposium on Theoretical Aspects of Computer Science Issue date: 2010 Date of publication: 09.03.2010

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