A new model of hard binary constraint Satisfaction Problems

Abstract

The phase transition for randomly generated binary Constraint Satisfaction Problems (CSPs) has recently been investigated by Smith[10], Smith & Dyer[11], and Prosser[8, 9]. It was found that in most cases one can accurately predict where the phase transition occurs using a predictor based on the expected number of solutions. Their results are based on a parameterization of CSPs that has a global constraint tightness value, that is, each constraint has the same tightness. In this paper we generalize their results using a parameterization of CSPs that has local constraint tightness values. We give a refined version of their predictor which incorporates the local graph topology of each individual CSP. It is shown that there is a similar phase transition in which constraint tightness does not have to be a global value and that the refined predictor better predicts the location of this phase transition. We also show that random problems generated with the refined predictor are as hard or harder to search than problems generated with the old predictor. Our results indicate that harder phase transitions can be found for NP-complete problems by generalizing the parameterizations used to model the problem in appropriate ways to include the structure of an individual problem.