Constraint Satisfaction Problems on Intervals and Lengths

Abstract

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. We study interval-valued constraint satisfaction problems (CSPs), in which the aim is to find an assignment of intervals to a given set of variables subject to constraints on the relative positions of intervals. Many well-known problems such as Interval Graph Recognition and Interval Satisfiability can be considered as examples of such CSPs. One interesting question concerning such problems is to determine exactly how the complexity of an interval-valued CSP depends on the set of constraints allowed in instances. For the framework known as Allen's interval algebra this question was completely answered earlier by the authors, by giving a complete description of the tractable cases and showing that all remaining cases are NP-complete.Here we extend the qualitative framework of Allen's algebra with additional constraints on the lengths of intervals. We allow these length constraints to be expressed as Horn disjunctive linear relations, a well-known tractable and sufficiently expressive form of constraints. The class of problems we consider contains, in particular, problems that are very closely related to the previously studied Unit Interval Graph Sandwich problem. We completely characterize sets of qualitative relations for which the CSP augmented with arbitrary length constraints of the above form is tractable. We also show that, again, all the remaining cases are NP-complete.

Tractability in constraint satisfaction problems: a survey

“…Allen's interval algebra consists of binary relations between intervals which are disjunctions of 13 basic interval relations (such as before, meets, includes, overlaps, starts, finishes or equals) [3]. For the problem of deciding whether there exist intervals on the real line satisfying a set of relations, a dichotomy has been given for all tractable subalgebras of Allen's algebra [125,126]. In the Region Containment Calculus (RCC-5) used in spatial reasoning, variables denote non-empty regions and the basic relations in the calculus express containment, disjointness, overlap or equality of pairs of regions.…”

The Complexity of Constraint Languages

“…This technique has been widely used in the analysis of Boolean constraint satisfaction problems [27,85], and in the analysis of temporal and spatial constraints [36,76,83,63,64]; it was introduced for the study of constraints over arbitrary finite sets in [49].…”

Design of robust fractional-order controllers and prefilters for multivariable system using interval constraint satisfaction technique

“…Interval Constraint Satisfaction Problems (ICSP) are combinatorial problems usually defined by a set of variables which are subject to a set of constraints and which take their values in a set of associated domains (intervals) [20,21]. Interval approach always guarantees numeric calculus and an inconsistency, which results in reliable solution of controller parameters.…”