The inverse k-max combinatorial optimization problem

Classical combinatorial optimization concerns finding a feasible subset of a ground set in order to optimize an objective function. We address in this article the inverse optimization problem with the k-max function. In other words, we attempt to perturb the weights of elements in the ground set at minimum total cost to make a predetermined subset optimal in the fashion of the k-max objective with respect to the perturbed weights. We first show that the problem is in general NP-hard. Regarding the case of independent feasible subsets, a combinatorial O(n2 log n) time algorithm is developed, where n is the number of elements in E. Special cases with improved complexity are also discussed.