Super Solutions in Constraint Programming

Where changes to a solution introduce additional expenses or reorganization, solution robustness is a valuable property. A robust solution is less sensitive to changes. For example, a robust schedule will not collapse immediately when one job takes slightly longer to execute than planned. The schedule should change locally and in small proportions, and the overall makespan should change little if at all. To improve solution robustness, we introduce the concept of super-solutions to constraint programming (CP). An (a, b)-super-solution is one in which if the values assigned to a variables are no longer available, the solution can be repaired by assigning these variables with a new values and at most b other variables. An (a, b)-super-solution is a generalization of both fault tolerant solutions in CP [14] and supermodels in propositional satisfiability (SAT) [9]. We show that finding (a, b)-super-solutions for any fixed a is NP-Complete in general. Super solutions are computed offline and do not require knowledge about the likely changes. A super-solution guarantees the existence of a small set of repairs when the future changes in a small way. In this paper, we focus on the algorithmic aspects of finding (1, 0)-supersolutions, which are the same as fault tolerant solutions [14]. A (1, 0)-supersolution is a solution where if one variable loses its value, we can find another solution by re-assigning this variable with a new value, and no other changes are required for the other variables. We explore methods based both on reformulation and on search to find (1, 0)-super-solutions. Our reformulation methods transform the constraint satisfaction problem so that the only solutions are super-solutions. We review two reformulation techniques presented in [14], and introduce a new one, which we call the cross-domain reformulation. Our search methods are based on notions of super-consistency. We propose two new search algorithms that extend the maintaining arc consistency algorithm (MAC [7, 6]). We empirically compare the different methods and observe that one of them, super-MAC shows considerable promise.