Guillaume Escamocher

Variable and value elimination in binary constraint satisfaction via forbidden patterns

Variable or value elimination in a constraint satisfaction problem (CSP) can be used in preprocessing or during search to reduce search space size. A variable elimination rule (value elimination rule) allows the polynomial-time identification of certain variables (domain elements) whose elimination, without the introduction of extra compensatory constraints, does not affect the satisfiability of an instance. We show that there are essentially just four variable elimination rules and three value elimination rules defined by forbidding generic sub-instances, known as irreducible existential patterns, in arc-consistent CSP instances. One of the variable elimination rules is the already-known Broken Triangle Property, whereas the other three are novel. The three value elimination rules can all be seen as strict generalisations of neighbourhood substitution.

Broken triangles

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 general-arity BTP-merging then allows us to extend the BTP tractable class previously defined only for binary CSP. We investigate the complexity of several related problems including the recognition problem for the general-arity BTP class when the variable order is unknown, finding an optimal order in which to apply BTP merges and detecting BTP-merges in the presence of global constraints such as AllDifferent.

Characterising the complexity of constraint satisfaction problems defined by 2-constraint forbidden patterns

Although the CSP (constraint satisfaction problem) is NP-complete, even in the case when all constraints are binary, certain classes of instances are tractable. We study classes of binary CSP instances defined by excluding subproblems. This approach has recently led to the discovery of novel tractable classes. The complete characterisation of all tractable classes defined by forbidding patterns (where a pattern is simply a compact representation of a set of subproblems) is a challenging problem. We demonstrate a dichotomy in the case of forbidden patterns consisting of either one or two constraints. This has allowed us to discover several new tractable classes including, for example, a novel generalisation of 2SAT. We then extend this dichotomy to existential patterns which are only forbidden on specific domain values.

On the Minimal Constraint Satisfaction Problem: Complexity and Generation

The Minimal Constraint Satisfaction Problem, or Minimal CSP for short, arises in a number of real-world applications, most notably in constraint-based product configuration. Despite its very permissive structure, it is NP-hard, even when bounding the size of the domains by

From Backdoor Key to Backdoor Completability: Improving a Known Measure of Hardness for the Satisfiable CSP

d\ge 9

Pushing the frontier of minimality

di?ź9. Yet very little is known about the Minimal CSP beyond that. Our contribution through this paper is twofold. Firstly, we generalize the complexity result to any value of d. We prove that the Minimal CSP remains NP-hard for

A dichotomy for 2-constraint forbidden CSP patterns

d\ge 3