K-adaptability in stochastic combinatorial optimization under objective uncertainty

Abstract

Abstract We address combinatorial optimization problems with uncertain objective functions, given by discrete probability distributions. Within this setting, we investigate the so-called K-adaptability approach: the aim is to calculate a set of K feasible solutions such that the objective value of the best of these solutions, calculated in each scenario independently, is optimal in expectation. Interpreted as a stochastic optimization problem, we only consider second-stage variables, however, the corresponding candidate solutions are selected in the first stage, i.e., before the scenario is known. We show that this problem is NP-hard even if the underlying certain problem is trivial, and present further complexity results concerning approximability and fixed-parameter tractability with respect to K. Moreover, we present exact solution methods as well as a heuristic for this problem and compare them in an extensive experimental evaluation, where the underlying problem is the Unconstrained Binary Optimization Problem, the Shortest Path Problem or the Spanning Tree Problem. It turns out that the performance and the ranking of these approaches strongly depends on the parameter K and on the number of scenarios.