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

TL;DR

This paper characterizes the computational complexity of constraint satisfaction problems over semilinear relations on the rationals and reals, introducing a polynomial-time algorithm for a new class of these problems.

Contribution

It generalizes known complexity results for semilinear CSPs, fully classifies their complexity for sets containing R+, and introduces an affine hull-based polynomial-time algorithm.

Findings

Polynomial-time algorithm for a new class of semilinear CSPs

Complete complexity classification for all finite sets containing R+

Analysis of linear optimization and integer solutions over these CSPs

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. 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.