Affine Consistency and the Complexity of Semilinear Constraints

Abstract

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.

Partially supported by the Swedish Research Council (VR) under grant 621-2012-3239.