Exact Phase Transitions in Random Constraint Satisfaction Problems

In this paper we propose a new type of random CSP model, called Model RB, which is a revision to the standard Model B. It is proved that phase transitions from a region where almost all problems are satis able to a region where almost all problems are unsatis able do exist for Model RB as the number of variables approaches in nity. Moreover, the critical values at which the phase transitions occur are also known exactly. By relating the hardness of Model RB to Model B, it is shown that there exist a lot of hard instances in Model RB.