When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2017.29
URN: urn:nbn:de:0030-drops-73846
URL: https://drops.dagstuhl.de/opus/volltexte/2017/7384/
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The Complexity of Holant Problems over Boolean Domain with Non-Negative Weights

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Abstract

Holant problem is a general framework to study the computational complexity of counting problems. We prove a complexity dichotomy theorem for Holant problems over the Boolean domain with non-negative weights. It is the first complete Holant dichotomy where constraint functions are not necessarily symmetric.

Holant problems are indeed read-twice #CSPs. Intuitively, some #CSPs that are #P-hard become tractable when restricted to read-twice instances. To capture them, we introduce the Block-rank-one condition. It turns out that the condition leads to a clear separation. If a function set F satisfies the condition, then F is of affine type or product type. Otherwise (a) Holant(F) is #P-hard; or (b) every function in F is a tensor product of functions of arity at most 2; or (c) F is transformable to a product type by some real orthogonal matrix. Holographic transformations play an important role in both the hardness proof and the characterization of tractability.

BibTeX - Entry

```@InProceedings{lin_et_al:LIPIcs:2017:7384,
author =	{Jiabao Lin and Hanpin Wang},
title =	{{The Complexity of Holant Problems over Boolean Domain with Non-Negative Weights}},
booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages =	{29:1--29:14},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-041-5},
ISSN =	{1868-8969},
year =	{2017},
volume =	{80},
editor =	{Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},