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

Peter Jonsson (1) , Johan Thapper (2)

1 Department of Computer and Information Science - Linköping University
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

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.

Mots clés: Constraint satisfaction problems; Semilinear sets; Algorithms; Computational complexity * Corresponding author

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]; Logique en informatique [cs.LO]

Peter Jonsson, Johan Thapper. Constraint Satisfaction and Semilinear Expansions of Addition over the Rationals and the Reals. Journal of Computer and System Sciences, 2016, 82 (5), pp.912 - 928. ⟨10.1016/j.jcss.2016.03.002⟩. ⟨hal-01796722⟩