Negative thinking in branch-and-bound: the case of unate covering

We introduce a new technique for solving some discrete optimization problems exactly. The motivation is that when searching the space of solutions by a standard branch-and-bound (B&B) technique, often a good solution is reached quickly and then improved only a few times before the optimum is found: hence, most of the solution space is explored to certify optimality, with no improvement in the cost function. This suggests that more powerful lower bounding would speed up the search dramatically. More radically, it would be desirable to modify the search strategy with the goal of proving that the given subproblem cannot yield a solution better than the current best one (negative thinking), instead of branching further in search for a better solution (positive thinking). For illustration we applied our approach to the unate covering problem. The algorithm starts in the positive-thinking mode by a standard B&B procedure that generates recursively smaller subproblems. If the current subproblem is "deep" enough, the algorithm switches to the negative thinking mode where it tries to prove that solving the subproblem does not improve the solution. The latter is achieved by a new search procedure invoked when the difference between the upper and lower bound is "small". Such a procedure is complete: either it yields a lower bound that matches the current upper bound, or it yields a new solution better than the current one. We implemented our new search procedure on top of ESPRESSO and SCHERZO, two state-of-art covering solvers used for computer-aided design applications, showing that in both cases we obtain new search engines (respectively, AURA and AURA II) much more efficient than the original ones.