Query-competitive algorithms for cheapest set problems under uncertainty

Considering the model of computing under uncertainty where element weights are uncertain but can be obtained at a cost by query operations, we study the problem of identifying a cheapest (minimum-weight) set among a given collection of feasible sets using a minimum number of queries of element weights. For the general case we present an algorithm that makes at most d⋅OPT+dd⋅OPT+d queries, where d is the maximum cardinality of any given set and OPT is the optimal number of queries needed to identify a cheapest set. For the minimum multi-cut problem in trees with d   terminal pairs, we give an algorithm that makes at most d⋅OPT+1d⋅OPT+1 queries. For the problem of computing a minimum-weight base of a given matroid, we give an algorithm that makes at most 2⋅OPT2⋅OPT queries, generalizing a known result for the minimum spanning tree problem. For each of the above algorithms we give matching lower bounds. We also settle the complexity of the verification version of the general cheapest set problem and the minimum multi-cut problem in trees under uncertainty.