Affine Consistency and the Complexity of Semilinear Constraints

Abstract

AbstractA 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.KeywordsPolynomial TimeConstraint SatisfactionConstraint Satisfaction ProblemAuxiliary ConditionUnary RelationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.