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A Dichotomy Theorem for the General Minimum Cost Homomorphism Problem

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Abstract

In the constraint satisfaction problem ($CSP$), the aim is to find an assignment of values to a set of variables subject to specified constraints. In the minimum cost homomorphism problem ($MinHom$), one is additionally given weights $c_{va}$ for every variable $v$ and value $a$, and the aim is to find an assignment $f$ to the variables that minimizes $\sum_{v} c_{vf(v)}$. Let $MinHom\left( \Gamma \right)$ denote the $MinHom$ problem parameterized by the set of predicates allowed for constraints. $MinHom\left( \Gamma \right)$ is related to many well-studied combinatorial optimization problems, and concrete applications can be found in, for instance, defence logistics and machine learning. We show that $MinHom\left( \Gamma \right)$ can be studied by using algebraic methods similar to those used for CSPs. With the aid of algebraic techniques, we classify the computational complexity of $MinHom\left( \Gamma \right)$ for all choices of $\Gamma$. Our result settles a general dichotomy conjecture previously resolved only for certain classes of directed graphs, [Gutin, Hell, Rafiey, Yeo, European J. of Combinatorics, 2008].

BibTeX - Entry

@InProceedings{takhanov:LIPIcs:2010:2493,
  author =	{Rustem Takhanov},
  title =	{{A Dichotomy Theorem for the General Minimum Cost Homomorphism Problem}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{657--668},
  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/2010/2493},
  URN =		{urn:nbn:de:0030-drops-24936},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2010.2493},
  annote =	{Keywords: Minimum cost homomorphisms problem, relational clones, constraint satisfaction problem, perfect graphs, supervised learning}
}

Keywords: Minimum cost homomorphisms problem, relational clones, constraint satisfaction problem, perfect graphs, supervised learning
Seminar: 27th International Symposium on Theoretical Aspects of Computer Science
Issue date: 2010
Date of publication: 2010


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