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Hard constraint satisfaction problems have hard gaps at location 1
Linköping University, Department of Computer and Information Science, TCSLAB - Theoretical Computer Science Laboratory. Linköping University, The Institute of Technology.
University of Durham.
Linköping University, Department of Computer and Information Science, TCSLAB - Theoretical Computer Science Laboratory. Linköping University, The Institute of Technology.
2009 (English)In: Theoretical Computer Science, ISSN 0304-3975, E-ISSN 1879-2294, Vol. 410, no 38-40, 3856-3874 p.Article in journal (Refereed) Published
Abstract [en]

An instance of the maximum constraint satisfaction problem (Max CSP) is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satisfied constraints. Max CSP captures many well-known problems (such as Maxk-SAT and Max Cut) and is consequently NP-hard. Thus, it is natural to study how restrictions on the allowed constraint types (or constraint language) affect the complexity and approximability of Max CSP. The PCP theorem is equivalent to the existence of a constraint language for which Max CSP has a hard gap at location 1; i.e. it is NP-hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable. All constraint languages, for which the CSP problem (i.e., the problem of deciding whether all constraints can be satisfied) is currently known to be NP-hard, have a certain algebraic property. We prove that any constraint language with this algebraic property makes Max CSP have a hard gap at location 1 which, in particular, implies that such problems cannot have a PTAS unless P = NP. We then apply this result to Max CSP restricted to a single constraint type; this class of problems contains, for instance, Max Cut and Max DiCut. Assuming P≠NP, we show that such problems do not admit PTAS except in some trivial cases. Our results hold even if the number of occurrences of each variable is bounded by a constant. Finally, we give some applications of our results.

Place, publisher, year, edition, pages
2009. Vol. 410, no 38-40, 3856-3874 p.
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:liu:diva-21225DOI: 10.1016/j.tcs.2009.05.022OAI: oai:DiVA.org:liu-21225DiVA: diva2:240973
Available from: 2009-09-30 Created: 2009-09-30 Last updated: 2017-12-13Bibliographically approved

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Jonsson, PeterKuivinen, Fredrik

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