Conjunctions of Among Constraints

Many existing global constraints can be encoded as a conjunction of among constraints. An among constraint holds if the number of the variables in its scope whose value belongs to a prespecified set, which we call its range, is within some given bounds. It is known that domain filtering algorithms can benefit from reasoning about the interaction of among constraints so that values can be filtered out taking into consideration several among constraints simultaneously. The present pa- per embarks into a systematic investigation on the circumstances under which it is possible to obtain efficient and complete domain filtering algorithms for conjunctions of among constraints. We start by observing that restrictions on both the scope and the range of the among constraints are necessary to obtain meaningful results. Then, we derive a domain flow-based filtering algorithm and present several applications. In particular, it is shown that the algorithm unifies and generalizes several previous existing results.Comment: 15 pages plus appendi

Representing Fitness Landscapes by Valued Constraints to Understand the Complexity of Local Search

Local search is widely used to solve combinatorial optimisation problems and to model biological evolution, but the performance of local search algorithms on different kinds of fitness landscapes is poorly understood. Here we introduce a natural approach to modelling fitness landscapes using valued constraints. This allows us to investigate minimal representations (normal forms) and to consider the effects of the structure of the constraint graph on the tractability of local search. First, we show that for fitness landscapes representable by binary Boolean valued constraints there is a minimal necessary constraint graph that can be easily computed. Second, we consider landscapes as equivalent if they allow the same (improving) local search moves; we show that a minimal normal form still exists, but is NP-hard to compute. Next we consider the complexity of local search on fitness landscapes modelled by valued constraints with restricted forms of constraint graph. In the binary Boolean case, we prove that a tree-structured constraint graph gives a tight quadratic bound on the number of improving moves made by any local search; hence, any landscape that can be represented by such a model will be tractable for local search. We build two families of examples to show that both the conditions in our tractability result are essential. With domain size three, even just a path of binary constraints can model a landscape with an exponentially long sequence of improving moves. With a treewidth two constraint graph, even with a maximum degree of three, binary Boolean constraints can model a landscape with an exponentially long sequence of improving moves