Negative thinking by incremental problem solving: application to unate covering

We introduce a new technique to solve exactly a discrete optimization problem, based on the paradigm of "negative" thinking. The motivation is that when searching the space of solutions, 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, but it does not yield any improvement of the cost function. So it is quite natural for an algorithm to be "skeptical" about the chance to improve the current best solution. For illustration we have applied our approach to the unate covering problem. We designed a procedure, raiser, implementing a negative thinking search, which is incorporated into a common branch-and-bound procedure. Experiments show that our program, AURA, outperforms both ESPRESSO and our enhancement of ESPRESSO using Coudert\´s limit lower bound. It is always faster and in the most difficult examples either has a running time better by up to two orders of magnitude, or the other programs fail to finish due to timeout or spaceout. The package SCHERZO is faster on some examples and loses on others, due to a less powerful pruning strategy of the search space, partially mitigated by a more effective computation of the maximal independent set.