Propagating Conjunctions of AllDifferent Constraints

TL;DR

This paper introduces a polynomial-time algorithm for propagating the conjunction of two AllDifferent constraints by leveraging a new theoretical extension of Hall's theorem, significantly improving efficiency in certain cases.

Contribution

It extends Hall's theorem to characterize simultaneous bipartite matchings and develops the first polynomial-time bound consistency algorithm for combined AllDifferent constraints.

Findings

Polynomial-time algorithm for convex bipartite graphs

Significant speedup on certain pathological problems

Experimental results show improved performance over existing methods

Abstract

We study propagation algorithms for the conjunction of two AllDifferent constraints. Solutions of an AllDifferent constraint can be seen as perfect matchings on the variable/value bipartite graph. Therefore, we investigate the problem of finding simultaneous bipartite matchings. We present an extension of the famous Hall theorem which characterizes when simultaneous bipartite matchings exists. Unfortunately, finding such matchings is NP-hard in general. However, we prove a surprising result that finding a simultaneous matching on a convex bipartite graph takes just polynomial time. Based on this theoretical result, we provide the first polynomial time bound consistency algorithm for the conjunction of two AllDifferent constraints. We identify a pathological problem on which this propagator is exponentially faster compared to existing propagators. Our experiments show that this new propagator can offer significant benefits over existing methods.