Short and Long Supports for Constraint Propagation

Abstract

Special-purpose constraint propagation algorithms frequently make implicit use of short supports -- by examining a subset of the variables, they can infer support (a justification that a variable-value pair may still form part of an assignment that satisfies the constraint) for all other variables and values and save substantial work -- but short supports have not been studied in their own right. The two main contributions of this paper are the identification of short supports as important for constraint propagation, and the introduction of HaggisGAC, an efficient and effective general purpose propagation algorithm for exploiting short supports. Given the complexity of HaggisGAC, we present it as an optimised version of a simpler algorithm ShortGAC. Although experiments demonstrate the efficiency of ShortGAC compared with other general-purpose propagation algorithms where a compact set of short supports is available, we show theoretically and experimentally that HaggisGAC is even better. We also find that HaggisGAC performs better than GAC-Schema on full-length supports. We also introduce a variant algorithm HaggisGAC-Stable, which is adapted to avoid work on backtracking and in some cases can be faster and have significant reductions in memory use. All the proposed algorithms are excellent for propagating disjunctions of constraints. In all experiments with disjunctions we found our algorithms to be faster than Constructive Or and GAC-Schema by at least an order of magnitude, and up to three orders of magnitude

Metamorphic Testing of Constraint Solvers

“…Any errors in Minion's implementation of the table constraint propagator would likely be found while testing it against all other constraints. Also, Minion includes several implementations of table constraints [15,20], which are compared. The testing process is automated in Python.…”

“…It is impossible to measure reliably the number of bugs it has discovered over time, as it has been used during the development of every constraint [13,14,20,15,19,11,12,9] added to Minion since 2007 and many bugs will be found at this stage. We can say no bug has been reported in any propagator which was published after development in Minion, except for one which was not added to the metamorphic tester.…”

Exploiting Short Supports for Improved Encoding of Arbitrary Constraints into SAT

“…For example, the short support {x 1 → 2, x 2 → 2, x 4 → 2, x 5 → 3} is a correct short support of size 4, but no literal can be removed from it without leaving at least one extension to a set of literals breaking the constraint. Following [19] we formally define short support as follows. Note from the definition that any full-length support is also a short support.…”

“…The idea of short support has already been successfully applied in constraint propagators [18,19,12]. Short support is defined above (Definition 1).…”

“…A short support of a constraint is similar to a satisfying tuple, but a short support is not required to assign every variable: some variables are left free to take any value. Where it is possible, exploiting short supports has proved to improve efficiency of GAC propagation [12,19]. We show that Bacchus's encoding of table constraints into SAT can be adapted to exploit short supports.…”