Random constraint satisfaction: Easy generation of hard (satisfiable) instances

Abstract

In this paper, we show that the models of random CSP instances proposed by Xu and Li [K. Xu, W. Li, Exact phase transitions in random constraint satisfaction problems, Journal of Artificial Intelligence Research 12 (2000) 93–103; K. Xu, W. Li, Many hard examples in exact phase transitions with application to generating hard satisfiable instances, Technical report, CoRR Report cs.CC/0302001, Revised version in Theoretical Computer Science 355 (2006) 291–302] are of theoretical and practical interest. Indeed, these models, called RB and RD, present several nice features. First, it is quite easy to generate random instances of any arity since no particular structure has to be integrated, or property enforced, in such instances. Then, the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness. In that case, a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold, i.e., to have an exponential tree-resolution complexity. Next, a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones. This analysis is supported by some representative results taken from an intensive experimentation that we have carried out, using complete and incomplete search methods.