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oai:drops-oai.dagstuhl.de:2493 2024-03-06T10:33:27Z ddc:004 open_access

A Dichotomy Theorem for the General Minimum Cost Homomorphism Problem Takhanov, Rustem Minimum cost homomorphisms problem relational clones constraint satisfaction problem perfect graphs supervised learning 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]. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Rustem Takhanov 2010 Is Part Of LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2010.2493 urn:nbn:de:0030-drops-24936 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2493 eng https://creativecommons.org/licenses/by-nd/3.0/legalcode