Section: New Results

Automata and Matrix Models

Participants : Nicolas Beldiceanu, Mats Carlsson, Pierre Flener, Justin Pearson.

Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables $ℳ$, with the same constraint $C$ defined by a finite-state automaton $𝒜$ on each row of $ℳ$ and a global cardinality constraint $\mathrm{𝑔𝑐𝑐}$ on each column of $ℳ$. We give two methods for deriving, by double counting, necessary conditions on the cardinality variables of the $\mathrm{𝑔𝑐𝑐}$ constraints from the automaton $𝒜$. The first method yields linear necessary conditions and simple arithmetic constraints. The second method introduces the cardinality automaton, which abstracts the overall behaviour of all the row automata and can be encoded by a set of linear constraints. We also provide a arc-consistency filtering algorithm for the conjunction of lexicographic ordering constraints between adjacent rows of $ℳ$ and (possibly different) automaton constraints on the rows. We evaluate the impact of our methods in terms of runtime and search effort on a large set of nurse rostering problem instances.

The corresponding paper On Matrices, Automata, and Double Counting in Constraint Programming [11] was published in the Constraints journal.