David A. Cohen

David A. Cohen, Peter G. Jeavons, Artem Kaznatcheev et al.

Local search in combinatorial optimisation can be viewed as an uphill climb on a corresponding fitness landscape, where the assignments visited by a strict local search follow an ascent in the landscape. This hill-climbing is sometimes surprisingly efficient, but not always. Since fitness landscapes can be succinctly represented by valued constraint satisfaction problems (VCSPs), it is natural to ask: what properties of VCSPs ensure that all ascents are polynomial? Or alternatively, what are the ``simplest'' VCSPs with exponential ascents? Prior examples of VCSPs with long ascents were built up as a chain of gadgets of constraints. Here we give a simpler star of gadgets construction by gluing 2n triangles of constraints at a common centre variable. We obtain a binary VCSP on 4n + 1 Boolean variables with an exponential ascent of length 10*2^n - 9. The variable at the centre of our construction intertwines two sublandscapes with only linear ascents into one with exponential ascents. The VCSP that we construct is significantly simpler than prior constructions in terms of treedepth (reducing Ω(log n) to 3) and feedback vertex set number (reducing Ω(n) to 1). We discuss the consequences of this simplicity for the parameterized complexity of local search.

David A. Cohen, Martin C. Cooper, Artem Kaznatcheev et al.

We investigate the complexity of local search based on steepest ascent. We show that even when all variables have domains of size two and the underlying constraint graph of variable interactions has bounded treewidth (in our construction, treewidth 7), there are fitness landscapes for which an exponential number of steps may be required to reach a local optimum. This is an improvement on prior recursive constructions of long steepest ascents, which we prove to need constraint graphs of unbounded treewidth.

Clement Carbonnel, David A. Cohen, Martin C. Cooper et al.

Singleton arc consistency is an important type of local consistency which has been recently shown to solve all constraint satisfaction problems (CSPs) over constraint languages of bounded width. We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints. We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT. Combined with simple counter-examples for other patterns, we make significant progress towards a complete classification.

David A. Cohen, Martin C. Cooper, Guillaume Escamocher et al.

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.

David A. Cohen, Peter G. Jeavons, Evgenij Thorstensen et al.

We study the complexity of constraint satisfaction problems involving global constraints, i.e., special-purpose constraints provided by a solver and represented implicitly by a parametrised algorithm. Such constraints are widely used; indeed, they are one of the key reasons for the success of constraint programming in solving real-world problems. Previous work has focused on the development of efficient propagators for individual constraints. In this paper, we identify a new tractable class of constraint problems involving global constraints of unbounded arity. To do so, we combine structural restrictions with the observation that some important types of global constraint do not distinguish between large classes of equivalent solutions.