Constraint Satisfaction Problems on Intervals and Length

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.

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