On inner independence systems

Abstract

A classic result of Korte and Hausmann [1978] and Jenkyns [1976] bounds the quality of the greedy solution to the problem of finding a maximum value basis of an independence system in terms of the rank‐quotient. We extend this result in two ways. First, we apply the greedy algorithm to an inner independence system contained in . Additionally, following an idea of Milgrom [2017], we incorporate exogenously given prior information about the set of likely candidates for an optimal basis in terms of a set . We provide a generalization of the rank‐quotient that yields a tight bound on the worst‐case performance of the greedy algorithm applied to the inner independence system relative to the optimal solution in . Furthermore, we show that for a worst‐case objective, the inner independence system approximation may outperform not only the standard greedy algorithm but also the inner matroid approximation proposed by Milgrom [2017]. Second, we generalize the inner approximation framework of independence systems to inner approximations of packing instances in by inner polymatroids and inner packing instances. We consider the problem of maximizing a separable discrete concave function and show that our inner approximation can be better than the greedy algorithm applied to the original packing instance. Our result provides a lower bound to the generalized rank‐quotient of a greedy algorithm to the optimal solution in this more general setting and subsumes Malinov and Kovalyov [1980]. We apply the inner approximation approach to packing instances induced by the FCC incentive auction and by two knapsack constraints.