Set Selection with Uncertain Weights: Non-Adaptive Queries and Thresholds

We study set selection problems where the weights are uncertain. Instead of its exact weight, only an uncertainty interval containing its true weight is available for each element. In some cases, some solutions are universally optimal; i.e., they are optimal for every weight that lies within the uncertainty intervals. However, it may be that no universal optimal solution exists, unless we are revealed additional information on the precise values of some elements. In the minimum cost admissible query problem, we are tasked to (non-adaptively) find a minimum-cost subset of elements that, no matter how they are revealed, guarantee the existence of a universally optimal solution. We introduce thresholds under uncertainty to analyze problems of minimum cost admissible queries. Roughly speaking, for every element e, there is a threshold for its weight, below which e is included in all optimal solutions and a second threshold above which e is excluded from all optimal solutions. We show that computing thresholds and finding minimum cost admissible queries are essentially equivalent problems. Thus, the analysis of the minimum admissible query problem reduces to the problem of computing thresholds. We provide efficient algorithms for computing thresholds in the settings of minimum spanning trees, matroids, and matchings in trees; and NP-hardness results in the settings of s-t shortest paths and bipartite matching. By making use of the equivalence between the two problems these results translate into efficient algorithms for minimum cost admissible queries in the settings of minimum spanning trees, matroids, and matchings in trees; and NP-hardness results in the settings of s-t shortest paths and bipartite matching.