Extending Broken Triangles and Enhanced Value-Merging

Abstract

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 17155The contribution was presented at CP 2016 :http://cp2016.a4cp.org/ Abstract. Broken triangles constitute an important concept not only for solving constraint satisfaction problems in polynomial time, but also for variable elimination or domain reduction by merging domain values. Specifically, for a given variable in a binary arc-consistent CSP, if no broken triangle occurs on any pair of values, then this variable can be eliminated while preserving satisfiability. More recently, it has been shown that even when this rule cannot be applied, it could be possible that for a given pair of values no broken triangle occurs. In this case, we can apply a domain-reduction operation which consists in merging these values while preserving satisfiability.In this paper we show that under certain conditions, and even if there are some broken triangles on a pair of values, these values can be merged without changing the satisfiability of the instance. This allows us to define a stronger merging operation and a new tractable class of binary CSP instances. We report experimental trials on benchmark instances.

Strengthening neighbourhood substitution

“…We use the notation b ik a for ∈ {i, k}. In Figure 1(a), the value 0 ∈ D(x 1 ) is snake substitutable by 1: we have 0 12 1 by taking e(1, 2, 1, 0) = 1…”

“…We illustrate the potential of SS, CNS and SCSS using the examples given in Figure 1. In Figure 1(a), the value 0 ∈ D(x 1 ) is snake substitutable by 1: we have 0 12 1 by taking e(1, 2, 1, 0) = 1 (where the arguments of e(i, k, a, d) are as shown in Figure 2), since (1, 1) ∈ R 12 and 0 23 − → 1; and 0 14 1 since 0 14 − → 1. Indeed, by a similar argument, the value 0 is snake substitutable by 1 in each domain.…”

“…For example, two values can be merged if the so-called broken triangle (BT) pattern does not occur on these two values [11]. Other valuemerging rules have been proposed which allow less merging than BT-merging but at a lower cost [24] or more merging at a greater cost [12,28]. Another family of satisfiability-preserving domain-reduction operations are based on the elimination of values that are not essential to obtain a solution [15].…”

Variable Elimination in Binary CSPs

“…The presence of some broken triangles on a given variable does not preclude value-merging or variable-elimination (while leaving satisfiability invariant): new solutions will not be introduced if the broken triangles lack support on some set of other variables (Cooper, El Mouelhi, & Terrioux, 2016b;El Mouelhi, 2018;Naanaa, 2016). Unfortunately, the search for lack-of-support variables for each broken triangle may render such techniques prohibitively expensive in terms of time complexity (since, in the worst case, the number of broken triangles is Θ(n 3 d 4 )).…”