Constraint Satisfaction and Semilinear Expansions of Addition over the Rationals and the Reals

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. We consider semilinear relations over the rationals and the reals. Under this assumption, the computational complexity of the constraint satisfaction problem (CSP) is known for all finite sets containing R+={(x,y,z) | x+y=z}, <=, and {1}. These problems correspond to expansions of the linear programming feasibility problem. We generalise this result and fully determine the complexity for all finite sets of semilinear relations containing R+. This is accomplished in part by introducing an algorithm, based on computing affine hulls, which solves a new class of semilinear CSPs in polynomial time. We further analyse the complexity of linear optimisation over the solution set and the existence of integer solutions.Comment: 22 pages, 1 figur

Affine Consistency and the Complexity of Semilinear Constraints

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

Affine consistency and the complexity of semilinear constraints

International audienceA 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