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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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