Statistical Analysis of Backtracking on Inconsistent CSPs

We analyze the distribution of computational eeort required by backtracking algorithms on unsatissable CSPs, using analogies with reliability models, where lifetime of a specimen before failure corresponds to the runtime of backtracking on unsatissable CSPs. We extend the results of 7] by showing empirically that the lognormal distribution is a good approximation of the backtracking eeort on unsolvable CSPs not only at the 50% satissable point, but in a relatively wide region. We also show how the law of proportionate eeect 9] commonly used to derive the lognormal distribution can be applied to modeling the number of nodes expanded in a search tree. Moreover, for certain intervals of C=N, where N is the number of variables, and C is the number of constraints, the parameters of the corresponding lognormal distribution can be approximated by the linear lognormal model 11] where mean log(deadends) is linear in C=N, and variance of log(deadends) is close to constant. The linear lognormal model allows us to extrapolate the results from a relatively easy overconstrained region to the hard critically constrained region and, in particular, to use more eecient strategies for testing backtracking algorithms .