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Affine Consistency and the Complexity of Semilinear Constraints
Linköping University, Department of Computer and Information Science, Software and Systems. Linköping University, The Institute of Technology. (TCSLAB)
Université Paris-Sud 11, Laboratoire de Recherche en Informatique (LRI) .
Number of Authors: 2
2014 (English)In: Mathematical Foundations of Computer Science 2014, Springer Berlin/Heidelberg, 2014, 420-431 p.Conference paper (Refereed)
Abstract [en]

A semilinear relation is a finite union of finite intersections of open and closed half-spaces over, for instance, the reals, the rationals or the integers. Semilinear relations have been studied in connection with algebraic geometry, automata theory, and spatiotemporal reasoning, just to mention a few examples. We concentrate on relations over the reals and rational numbers. Under this assumption, the computational complexity of the constraint satisfaction problem (CSP) is known for all finite sets Γ of semilinear relations containing the relations R +={(x,y,z) | x+y=z}, ≤ and {1}. These problems correspond to extensions of LP feasibility. We generalise this result as follows. We introduce an algorithm, based on computing affine hulls, which solves a new class of semilinear CSPs in polynomial time. This allows us to fully determine the complexity of CSP(Γ) for semilinear Γ containing R+ and satisfying two auxiliary conditions. Our result covers all semilinear Γ such that {R+,{1}}⊆Γ. We continue by studying the more general case when Γ contains R+ but violates either of the two auxiliary conditions. We show that each such problem is equivalent to a problem in which the relations are finite unions of homogeneous linear sets and we present evidence that determining the complexity of these problems may be highly non-trivial.

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2014. 420-431 p.
Lecture Notes in Computer Science, ISSN 0302-9743 (print), 1611-3349 (online)
National Category
Computer and Information Science
URN: urn:nbn:se:liu:diva-112904DOI: 10.1007/978-3-662-44465-8_36ISI: 000358254600036ScopusID: 2-s2.0-84906239762ISBN: 978-3-662-44464-1OAI: diva2:773660
39th International Symposium on Mathematical Foundations of Computer Science (MFCS-2014)
Available from: 2014-12-19 Created: 2014-12-19 Last updated: 2015-08-20

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