The Brélaz Heuristic and Optimal Static Orderings

The order in which the variables are assigned can have an enormous impact on the time taken by a backtracking search algorithm to solve a constraint satisfaction problem (CSP). The Br elaz heuristic is a dynamic variable ordering heuristic which has been shown to give good results for some classes of binary CSPs when the constraint graph is not complete. Its advantage over the simpler smallest-domain heuristic is that it uses information about the constraint graph. This paper uses theoretical work by Nudel to assess the performance of the Br elaz heuristic. Nudel's work gives the expected number of nodes at each level of the search tree when using the forward checking algorithm to nd all solutions to a CSP, given a speci ed order of the variables. From this, optimal static orderings are found for a sample of small binary CSPs. The optimal orderings are used to learn rules for a static ordering heuristic, which are converted into modi cations to the Br elaz heuristic. The improved heuristic is shown to halve the mean search cost of solving sparse random binary CSPs with 50 variables, at the phase transition. However, our modi cations, and the Br elaz heuristic itself, are mainly in the form of improved tie-breakers for the smallest-domain heuristic, which the results suggest is still the basis of good heuristics for this class of problem.