Achieving Domain Consistency and Counting Solutions for Dispersion Constraints

Abstract

Many combinatorial problems require of their solutions that they achieve a certain balance of given features. For this important aspect of modeling, the spread and deviation constraints have been proposed in Constraint Programming to express balance among a set of variables by constraining their mean and their overall deviation from the mean. Currently the only practical filtering algorithms known for these constraints achieve bounds consistency. In this paper we improve that filtering by presenting an efficient domain consistency algorithm that applies to both constraints. We also extend it to count solutions so that it can be used in counting-based search, a generic and effective family of branching heuristics that free the user from having to write problem-specific search heuristics. We provide a time complexity analysis of our contributions and also evaluate them empirically on benchmark problems.

A parametric propagator for pairs of Sum constraints with a discrete convexity property

“…Finally, it should be noted that there are domain-consistent propagators for several constraints for which our propagator achieves bounds(Z) consistency or bounds(R) consistency only. In particular, Trick [24] presented a domainconsistent propagator for Linear = and Pesant [25] presented a domain-consistent propagator for L p -Norm (including Spread and Deviation). Those propagators are based on dynamic programming ideas and have a higher time complexity than our approach.…”

The BIN_COUNTS Constraint: Filtering and Applications

“…The objective that is optimised in order to achieve a balanced curriculum varies from work to work: the original formulation seeks to minimise the maximum load in any given semester, other formulations, see e.g. [24], employ the L 2 -deviation to measure balance. In this section we discuss a different strategy, based on the χ 2 test statistic, to measure curriculum compliance with a target credit load distribution among semesters.…”

“…(1) global cardinality 1,...,S (s; c) (2) bin packing w (s; l) (3) s i < s j (course prerequisites) (4) bin counts b1,...,bm+1 (l; o) As discussed in [24], a CP formulation for the BACP (Fig. 8) includes one decision variable s i per course i, which indicates to which semester the course is assigned; one decision variables l j per semester j recording its academic load; and one decision variable c j per semester recording the number of courses allocated to it.…”

“…CP approaches based on the spread constraint were discussed in [39,37], where the authors proposed a decomposition strategy that pre-computes the number of nurses for each zone and then solves each zone separately. An alternative CP approach based on the dispersion constraint was discussed in [24] and more recently in [25], where nurse dependent workloads are modelled.…”

From Support Propagation to Belief Propagation in Constraint Programming

“…Valiant, 1979). Nevertheless the recent work on counting-based search has provided efficient algorithms to count either exactly or approximately for several constraints (Pesant et al, 2012;Brockbank, Pesant, & Rousseau, 2013;Pesant, 2015). An efficient implementation of belief propagation with global constraints needs to perform counting weighted by the beliefs about variables: this is close in spirit to other work on counting for optimization constraints in which individual costs are associated with each variable assignment (Pesant, 2016;Delaite & Pesant, 2017).…”