Range and Roots: Two common patterns for specifying and propagating counting and occurrence constraints

Abstract

We propose Range and Roots which are two common patterns useful for specifying a wide range of counting and occurrence constraints. We design specialised propagation algorithms for these two patterns. Counting and occurrence constraints specified using these patterns thus directly inherit a propagation algorithm. To illustrate the capabilities of the Range and Roots constraints, we specify a number of global constraints taken from the literature. Preliminary experiments demonstrate that propagating counting and occurrence constraints using these two patterns leads to a small loss in performance when compared to specialised global constraints and is competitive with alternative decompositions using elementary constraints. * This paper is a compilation and an extension of [10], [11], and [12]. The first author was supported by the ANR project ANR-06-BLAN-0383-02.

Estimating the Number of Solutions of Cardinality Constraints Through $$\texttt {range}$$ and $$\texttt {roots}$$ Decompositions

“…The range and roots constraints [2] are two auxiliary constraints that can help decomposing a lot of cardinality constraints. In this study, we will use these decomposition to count solutions on cardinality constraints.…”

“…In this study, we will use these decomposition to count solutions on cardinality constraints. As the authors wrote in [2], "range captures the notion of image of a function and roots captures the notion of domain". In this paper, we use alternative definitions for these constraints, equivalent to those of [2] and better suited to our needs.…”

“…These coefficients are referenced as the "triangles of numbers" in OEIS. 1 The coefficients a n,m corresponds to the number of possible surjections from a set of cardinal n into a set of cardinal m. 2 There is a non-recursive formula to compute these coefficients. The following results is admitted here.…”

“…In [2], the authors introduce two new global constraints range and roots, that can be used to specify many cardinality constraints. In other words, for almost every cardinality constraint, there is an equivalent model using only the more primitive range and roots constraints (and some arithmetic constraints).…”

Conjunctions of Among Constraints

“…Other approaches to the design of filtering algorithms for combinations (but not necessarily conjunctions) of global (but not necessarily among) constraints are described in [4,7,8]. The method introduced in [4] deals with logical combinations of some primitive constraints but differs substantially from ours in the sense that it cannot capture a single among constraint.…”