On Broken Triangles

Abstract

A binary CSP instance satisfying the broken-triangle property (BTP) can be solved in polynomial time. Unfortunately, in practice, few instances satisfy the BTP. We show that a local version of the BTP allows the merging of domain values in arbitrary instances of binary CSP, thus providing a novel polynomial-time reduction operation. Extensive experimental trials on benchmark instances demonstrate a significant decrease in instance size for certain classes of problems. We show that BTP-merging can be generalised to instances with constraints of arbitrary arity and we investigate the theoretical relationship with resolution in SAT. A directional version of the general-arity BTP then allows us to extend the BTP tractable class previously defined only for binary CSP.

Extending Broken Triangles and Enhanced Value-Merging

“…The 1-wBTP-merging algorithm is similar to the algorithm for BTPmerging [5]. More specifically, given a variable x k , we check for each pair of values v…”

“…Even when this variable-elimination rule cannot be applied, it can nevertheless happen that no broken triangle occurs on a particular pair of values. In this case, these two values can be merged into a single value without changing the satisfiability of the instance [5]. This domain-reduction operation, known as BT-merging, was found to be applicable in diverse benchmark domains, although extensive experimental trials would seem to indicate that it is not useful, in terms of total solving time, as a preprocessing operation in a general-purpose solver [4].…”

“…We go on to show, in Sect. 5, that this merging rule does not allow the elimination of variables. Nevertheless, in Sect.…”

Broken Triangles Revisited

“…The notion of broken triangle has generated a certain amount of interest in the constraints community: it has led to the definition of novel tractable classes [3,7], variable elimination rules [1] and domain reduction rules [4,5]. The merging of pairs of values in the same variable domain which do not belong to a broken triangle has been shown to lead to considerable reduction of search space size for certain benchmark instances of binary CSP [4].…”

“…The merging of pairs of values in the same variable domain which do not belong to a broken triangle has been shown to lead to considerable reduction of search space size for certain benchmark instances of binary CSP [4]. The corresponding reduction operation, known as BTP-merging, is satisfiability-preserving and is therefore worthy of a deeper theoretical analysis as a potentially useful preprocessing operation.…”