A Branch and Bound Algorithm for Numerical MAX-CSP

Abstract

The Constraint Satisfaction Problem (CSP) framework allows users to define problems in a declarative way, quite independently from the solving process. However, when the problem is over-constrained, the answer “no solution” is generally unsatisfactory. A Max-CSP $\mathcal{P}_m = \langle V, \textbf{D}, C \rangle$ is a triple defining a constraint problem whose solutions maximise constraint satisfaction. In this paper, we focus on numerical CSPs, which are defined on real variables represented as floating point intervals and which constraints are numerical relations defined in extension. Solving such a problem (i.e., exactly characterizing its solution set) is generally undecidable and thus consists in providing approximations. We propose a branch and bound algorithm that computes under and over approximations of its solution set and determines the maximum number of satisfied constraints. The technique is applied on three numeric applications and provides promising results.