License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2018.12
URN: urn:nbn:de:0030-drops-90168
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9016/
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Backens, Miriam

A Complete Dichotomy for Complex-Valued Holant^c

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LIPIcs-ICALP-2018-12.pdf (0.5 MB)

Abstract

Holant problems are a family of counting problems on graphs, parametrised by sets of complex-valued functions of Boolean inputs. Holant^c denotes a subfamily of those problems, where any function set considered must contain the two unary functions pinning inputs to values 0 or 1. The complexity classification of Holant problems usually takes the form of dichotomy theorems, showing that for any set of functions in the family, the problem is either #P-hard or it can be solved in polynomial time. Previous such results include a dichotomy for real-valued Holant^c and one for Holant^c with complex symmetric functions, i.e. functions which only depend on the Hamming weight of the input.
Here, we derive a dichotomy theorem for Holant^c with complex-valued, not necessarily symmetric functions. The tractable cases are the complex-valued generalisations of the tractable cases of the real-valued Holant^c dichotomy. The proof uses results from quantum information theory, particularly about entanglement. This full dichotomy for Holant^c answers a question that has been open for almost a decade.

BibTeX - Entry

@InProceedings{backens:LIPIcs:2018:9016,
  author =	{Miriam Backens},
  title =	{{A Complete Dichotomy for Complex-Valued Holant^c}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{12:1--12:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9016},
  URN =		{urn:nbn:de:0030-drops-90168},
  doi =		{10.4230/LIPIcs.ICALP.2018.12},
  annote =	{Keywords: computational complexity, counting complexity, Holant problems, dichotomy, entanglement}
}

Keywords: computational complexity, counting complexity, Holant problems, dichotomy, entanglement
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018

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