Optimizing Backtrack Search for all Solutions to Conjunctive Problems

We consider the problem of minimizing depth-first search effort for the generation of all solutions to a problem stated as a conjunction of subproblems. For a sequence of subproblems that share no variables, the effort is minimized by ordering the subproblems in decreasing ratio of NC/(N 1), where N and C are the number of solutions and the search effort of obtaining a solution to the subproblem. If a conjunctive problem has an arbitrary number of subproblems sharing variables among them, we assume that in the solution sets of the subproblems, each argument variable is bound to elements of its domain with equal frequency. Under this uniform distribution assumption, we derive a set of necessary conditions that must be satisfied by an optimal depth-first sequence. If the distribution assumption does not hold for a conjunctive problem, then the search effort can be optimized only if the sequencing of subproblems is suitably interleaved with the actual enumeration of solutions to the problem.