License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2021.27
URN: urn:nbn:de:0030-drops-144679
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14467/
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Butti, Silvia ; Dalmau, Víctor

Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem

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LIPIcs-MFCS-2021-27.pdf (0.9 MB)

Abstract

Given a pair of graphs ? and ?, the problems of deciding whether there exists either a homomorphism or an isomorphism from ? to ? have received a lot of attention. While graph homomorphism is known to be NP-complete, the complexity of the graph isomorphism problem is not fully understood. A well-known combinatorial heuristic for graph isomorphism is the Weisfeiler-Leman test together with its higher order variants. On the other hand, both problems can be reformulated as integer programs and various LP methods can be applied to obtain high-quality relaxations that can still be solved efficiently. We study so-called fractional relaxations of these programs in the more general context where ? and ? are not graphs but arbitrary relational structures. We give a combinatorial characterization of the Sherali-Adams hierarchy applied to the homomorphism problem in terms of fractional isomorphism. Collaterally, we also extend a number of known results from graph theory to give a characterization of the notion of fractional isomorphism for relational structures in terms of the Weisfeiler-Leman test, equitable partitions, and counting homomorphisms from trees. As a result, we obtain a description of the families of CSPs that are closed under Weisfeiler-Leman invariance in terms of their polymorphisms as well as decidability by the first level of the Sherali-Adams hierarchy.

BibTeX - Entry

@InProceedings{butti_et_al:LIPIcs.MFCS.2021.27,
  author =	{Butti, Silvia and Dalmau, V{\'\i}ctor},
  title =	{{Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{27:1--27:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14467},
  URN =		{urn:nbn:de:0030-drops-144679},
  doi =		{10.4230/LIPIcs.MFCS.2021.27},
  annote =	{Keywords: Weisfeiler-Leman algorithm, Sherali-Adams hierarchy, Graph homomorphism, Constraint Satisfaction Problem}
}

Keywords: Weisfeiler-Leman algorithm, Sherali-Adams hierarchy, Graph homomorphism, Constraint Satisfaction Problem
Collection: 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)
Issue Date: 2021
Date of publication: 18.08.2021

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