On Matrices, Automata, and Double Counting

Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables M, with the same constraint defined by a finite-state automaton A on each row of M and a global cardinality constraint gcc on each column of M. We give two methods for deriving, by double counting, necessary conditions on the cardinality variables of the 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 evaluate the impact of our methods on a large set of nurse rostering problem instances

Revisiting the tree Constraint

International audienceThis paper revisits the tree constraint introduced in [2] which partitions the nodes of a n-nodes, m-arcs directed graph into a set of node-disjoint anti-arborescences for which only certain nodes can be tree roots. We introduce a new filtering algorithm that enforces generalized arc-consistency in O(n + m) time while the original filtering algorithm reaches O(nm) time. This result allows to tackle larger scale problems involving graph partitioning

Generating Linear Invariants for a Conjunction of Automata Constraints

International audienceWe propose a systematic approach for generating linear implied constraints that link the values returned by several automata with accumulators after consuming the same input sequence. The method handles automata whose accumulators are increased by (or reset to) some non-negative integer value on each transition. We evaluate the impact of the generated linear invariants on conjunctions of two families of time-series constraints

A Lagrangian Relaxation for Golomb Rulers

Abstract. The Golomb Ruler Problem asks to position n integer marks on a ruler such that all pairwise distances between the marks are distinct and the ruler has minimum total length. It is a very challenging combinatorial problem, and provably optimal rulers are only known for n up to 26. Lower bounds can be obtained using Linear Programming formulations, but these are computationally expensive for large n. In this paper, we propose a new method for finding lower bounds based on a Lagrangian relaxation. We present a combinatorial algorithm that finds good bounds quickly without the use of a Linear Programming solver. This allows us to embed our algorithm into a constraint programming search procedure. We compare our relaxation with other lower bounds from the literature, both formally and experimentally. We also show that our relaxation can reduce the constraint programming search tree considerably.

Solving Nurse Rostering Problems Using Soft Global Constraints

International audienceNurse Rostering Problems (NRPs) consist of generating rosters where required shifts are assigned to nurses over a scheduling period satisfying a number of constraints. Most NRPs in real world are NP-hard and are particularly challenging as a large set of different constraints and specific nurse preferences need to be satisfied. The aim of this paper is to show how NRPs can be easily modelled and efficiently solved using soft global constraints. Experiments on real-life problems and comparison with ad'hoc OR approaches are detailed