Takhanov, Rustem
A Dichotomy Theorem for the General Minimum Cost Homomorphism Problem
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 wellstudied 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 = {657668},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897163},
ISSN = {18688969},
year = {2010},
volume = {5},
editor = {JeanYves Marion and Thomas Schwentick},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2010/2493},
URN = {urn:nbn:de:0030drops24936},
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 