Scaling Effects in the CSP Phase Transition

Phase transitions in constraint satisfaction problems (CSP's) are the subject of intense study. We identify an order parameter for random binary CSP's. There is a rapid transition in the probability of a CSP having a solution at a critical value of this parameter. The order parameter allows diierent phase transition behaviour to be compared in an uniform manner, for example CSP's generated under diierent regimes. We then show that within classes, the scaling of behaviour can be modelled by a tehnique called \\nite size scaling". This applies not only to probability of solu-bility, as has been observed before in other NP-problems, but also to search cost, the rst time this has been observed. Furthermore, the technique applies with equal validity to several diierent methods of varying problem size. As well as contributing to the understanding of phase transitions, we contribute by allowing much ner grained comparison of algorithms, and for accurate empirical extrapolations of behaviour.