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Computational Complexity of the Minimum Cost Homomorphism Problem on Three-element Domains
Linköping University, Department of Computer and Information Science, Software and Systems. Linköping University, The Institute of Technology. (TCSLAB)
Number of Authors: 1
2014 (English)Conference paper (Refereed)
Abstract [en]

In this paper we study the computational complexity of the extended minimum cost homomorphism problem (Min-Cost-Hom) as a function of a constraint language, i.e. a set of constraint relations and cost functions that are allowed to appear in instances. A wide range of natural combinatorial optimisation problems can be expressed as extended Min-Cost-Homs and a classification of their complexity would be highly desirable, both from a direct, applied point of view as well as from a theoretical perspective.

The extended Min-Cost-Hom can be understood either as a flexible optimisation version of the constraint satisfaction problem (CSP) or a restriction of the (general-valued) valued constraint satisfaction problem (VCSP). Other optimisation versions of CSPs such as the minimum solution problem (Min-Sol) and the minimum ones problem (Min-Ones) are special cases of the extended Min-Cost-Hom.

The study of VCSPs has recently seen remarkable progress. A complete classification for the complexity of finite-valued languages on arbitrary finite domains has been obtained Thapper and Živný [STOC’13]. However, understanding the complexity of languages that are not finitevalued appears to be more difficult. The extended Min-Cost-Hom allows us to study problematic languages of this type without having to deal with with the full generality of the VCSP. A recent classification for the complexity of three-element Min-Sol, Uppman [ICALP’13], takes a step in this direction. In this paper we generalise this result considerably by determining the complexity of three-element extended Min-Cost-Hom.

Place, publisher, year, edition, pages
Dagstuhl: Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik , 2014. 651-662 p.
, Leibniz International Proceedings in Informatics, ISSN 1868-8969 ; 25
National Category
Computer Science
URN: urn:nbn:se:liu:diva-112916DOI: 10.4230/LIPIcs.STACS.2014.651ISBN: 978-3-939897-65-1OAI: diva2:773698
31st International Symposium on Theoretical Aspects of Computer Science (STACS-2014)
Available from: 2014-12-19 Created: 2014-12-19 Last updated: 2015-04-23
In thesis
1. On Some Combinatorial Optimization Problems: Algorithms and Complexity
Open this publication in new window or tab >>On Some Combinatorial Optimization Problems: Algorithms and Complexity
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is about the computational complexity of several classes of combinatorial optimization problems, all related to the constraint satisfaction problems.

A constraint language consists of a domain and a set of relations on the domain. For each such language there is a constraint satisfaction problem (CSP). In this problem we are given a set of variables and a collection of constraints, each of which is constraining some variables with a relation in the language. The goal is to determine if domain values can be assigned to the variables in a way that satisfies all constraints. An important question is for which constraint languages the corresponding CSP can be solved in polynomial time. We study this kind of question for optimization problems related to the CSPs.

The main focus is on extended minimum cost homomorphism problems. These are optimization versions of CSPs where instances come with an objective function given by a weighted sum of unary cost functions, and where the goal is not only to determine if a solution exists, but to find one of minimum cost. We prove a complete classification of the complexity for these problems on three-element domains. We also obtain a classification for the so-called conservative case.

Another class of combinatorial optimization problems are the surjective maximum CSPs. These problems are variants of CSPs where a non-negative weight is attached to each constraint, and the objective is to find a surjective mapping of the variables to values that maximizes the weighted sum of satisfied constraints. The surjectivity requirement causes these problems to behave quite different from for example the minimum cost homomorphism problems, and many powerful techniques are not applicable. We prove a dichotomy for the complexity of the problems in this class on two-element domains. An essential ingredient in the proof is an algorithm that solves a generalized version of the minimum cut problem. This algorithm might be of independent interest.

In a final part we study properties of NP-hard optimization problems. This is done with the aid of restricted forms of polynomial-time reductions that for example preserves solvability in sub-exponential time. Two classes of optimization problems similar to those discussed above are considered, and for both we obtain what may be called an easiest NP-hard problem. We also establish some connections to the exponential time hypothesis.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. 32 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1663
Computational complexity, optimization, constraint satisfaction problem
National Category
Computer Science
urn:nbn:se:liu:diva-116859 (URN)10.3384/diss.diva-116859 (DOI)978-91-7519-072-3 (print) (ISBN)
Public defence
2015-05-21, Alan Turing, E-huset, Campus Valla, Linköping, 13:15 (English)
CUGS (National Graduate School in Computer Science), 09.01
Available from: 2015-04-23 Created: 2015-04-07 Last updated: 2015-04-27Bibliographically approved

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