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A semilinear relation is a finite union of finite intersections of open andclosed half-spaces over, for instance, the reals, the rationals, or theintegers. Semilinear relations have been studied in connection with algebraicgeometry, automata theory, and spatiotemporal reasoning. We consider semilinearrelations over the rationals and the reals. Under this assumption, thecomputational complexity of the constraint satisfaction problem (CSP) is knownfor all finite sets containing R+={(x,y,z) | x+y=z}, <=, and {1}. Theseproblems correspond to expansions of the linear programming feasibilityproblem. We generalise this result and fully determine the complexity for allfinite sets of semilinear relations containing R+. This is accomplished in partby introducing an algorithm, based on computing affine hulls, which solves anew class of semilinear CSPs in polynomial time. We further analyse thecomplexity of linear optimisation over the solution set and the existence ofinteger solutions.

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