On Matrices, Automata, and Double Counting

Abstract

Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables M, with the same constraint defined by a finitestate 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.

Solution neighbourhoods for constraint-directed local search

“…In Table 1, we give a comparison of the two search procedures in Sections 4.2.1 and 4.2.2 on NSPlib instances [7], except for those known from [2] to be unsatisfiable (in order to accelerate the experiments). Other instances might be unsatisfiable (this information cannot be extracted from the NSPlib website), hence the percentage of solved instances might not reach 100% for that reason.…”

“…For each run of an instance, all search procedures started with the same randomly generated initial assignment. We did not run instances of cases 1 to 10 and cases 12 to 14, as they are very easy [2]; we also did not run instances of case 11, as the DFAs (30 or 60 DFAs with 11816 states and 41922 transitions each, before unrolling) for these instances are too large for handling by the automaton constraint of [4] under Comet (version 2.1.1) on the chosen hardware. Note that this memory problem is independent of search, hence both our search procedures in Sections 4.2.1 and 4.2.2 suffer from the same memory problem.…”

“…For some other constraints, there are ways to design a solution neighbourhood. For example, consi- Table 1: Benchmark results over 25 runs each on NSPlib instances, except for those known from [2] to be unsatisfiable (in order to accelerate the experiments). Other instances might be unsatisfiable (this information cannot be extracted from the NSPlib website), hence %S (the average percentage of instances solved without timing out) might not reach 100% for that reason.…”

On matrices, automata, and double counting in constraint programming

“…The tractability of propagating the matrix-of-automaton-and-gcc pattern of our [2] and the present extension thereof is studied in [14]. …”

“…, R}. Each row r (with 0 ≤ r < R) of M is subject to a constraint defined by an automaton 2 A and, depending on the search procedure, we may break symmetries by a lexicographic ordering between adjacent rows [7,11,12]. For simplicity (except in Section 5), we assume that each row is subject to the same constraint.…”