Approximating the min-max (regret) selecting items problem

In this paper the problem of selecting p items out of n available to minimize the total cost is discussed. This problem is a special case of many important combinatorial optimization problems such as 0-1 knapsack, minimum assignment, single machine scheduling, minimum matroid base or resource allocation. It is assumed that the item costs are uncertain and they are specified as a scenario set containing K distinct cost scenarios. In order to choose a solution the min-max and min-max regret criteria are applied. It is shown that both min-max and min-max regret problems are not approximable within any constant factor unless P = NP, which strengthens the results known up to date. In this paper a deterministic approximation algorithm with performance ratio of O(ln K) for the min-max version of the problem is also proposed. (c) 2012 Elsevier B.V. All rights reserved.