Phase transition behaviour in constraint satisfaction problems

Abstract

Many problems in artificial intelligence and computer science can be formulated as constraint satisfaction problems (CSPs). A CSP consists of a set of variables among which a set of constraints are imposed, with a solution corresponding to an assignment for every variable such that no constraints are violated. Most forms of CSP are NP-complete. Recent research has shown that the CSP exhibits a phase transition as a control parameter is varied. This transition lies between a region where most problems are easy and soluble, and a region where most problems are easy but insoluble. In the intervening phase transition region, the average problem difficulty is greatest. Phase transition behaviour can be exploited to create test beds of hard and easy problems for CSP algorithms. In this thesis, we study the phase transition of the binary CSP and examine various aspects of complete search algorithms for it. The phenomenon of exceptionally hard problems (`ehps') is examined in detail: these are rare searches on easy problems which become exceptionally expensive for a particular complete algorithm following a poor early search move. An explanation for the occurrence of ehps is proposed, and the relative susceptibility of certain algorithms to the phenomenon is explored. We then show that the phase transition paradigm can be applied to two tasks of polynomial cost complexity: attempting to establish arc and path consistency in a CSP. Phase transition behaviour analogous to that found when searching for a solution is demonstrated for these tasks, and the effectiveness and cost of establishing arc and path consistency is examined. The theme of establishing consistency in CSPs is extended by studying an algorithm which maintains arc consistency during search. Its performance is compared with that of an algorithm which maintains a lower level of consistency, and it is shown that the higher level of consistency reduces average search cost and ehp behaviour on many types of CSP. Finally, the subject of dynamically selecting the variable to instantiate at each stage in the search process is considered. We compare a number of heuristics which attempt to select the variable most likely to lead to failure, and show that the supposed principle behind these appears to be fundamentally flawed