The RegularGcc Matrix Constraint

TL;DR

This paper investigates the propagation complexity of the RegularGcc matrix constraint, proving NP-hardness in general but identifying fixed parameter tractable cases and proposing improved propagation methods with experimental validation.

Contribution

It introduces new tractability results for RegularGcc propagation, identifies necessary conditions for solutions, and enhances propagation efficiency with weighted automata.

Findings

Propagation is NP-hard under strong restrictions.

Two cases where propagation is fixed parameter tractable.

Improved propagation methods outperform simple decompositions.

Abstract

We study propagation of the RegularGcc global constraint. This ensures that each row of a matrix of decision variables satisfies a Regular constraint, and each column satisfies a Gcc constraint. On the negative side, we prove that propagation is NP-hard even under some strong restrictions (e.g. just 3 values, just 4 states in the automaton, or just 5 columns to the matrix). On the positive side, we identify two cases where propagation is fixed parameter tractable. In addition, we show how to improve propagation over a simple decomposition into separate Regular and Gcc constraints by identifying some necessary but insufficient conditions for a solution. We enforce these conditions with some additional weighted row automata. Experimental results demonstrate the potential of these methods on some standard benchmark problems.