In this paper the minimum spanning tree problem with uncertain edge costs is discussed. In order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution. The min-max, min-max regret and 2-stage min-max versions of the problem are discussed. The complexity and approximability of all these problems are explored. It is proved that the min-max and min-max regret versions with nonnegative edge costs are hard to approximate within O(log 1−ǫ n) for any ǫ > 0 unless the problems in NP have quasi-polynomial time algorithms. Similarly, the 2-stage min-max problem cannot be approximated within O(log n) unless the problems in NP have quasi-polynomial time algorithms. In this paper randomized LP-based approximation algorithms with performance ratio of O(log 2 n) for min-max and 2-stage min-max problems are also proposed.
In this paper the problem of selecting p out of n available items is discussed, such that their total cost is minimized. We assume that costs are not known exactly, but stem from a set of possible outcomes.Robust recoverable and two-stage models of this selection problem are analyzed. In the two-stage problem, up to p items is chosen in the first stage, and the solution is completed once the scenario becomes revealed in the second stage. In the recoverable problem, a set of p items is selected in the first stage, and can be modified by exchanging up to k items in the second stage, after a scenario reveals.We assume that uncertain costs are modeled through bounded uncertainty sets, i.e., the interval uncertainty sets with an additional linear (budget) constraint, in their discrete and continuous variants. Polynomial algorithms for recoverable and two-stage selection problems with continuous bounded uncertainty, and compact mixed integer formulations in the case of discrete bounded uncertainty are constructed.
In this paper the following selection problem is discussed. A set of n items is given and we wish to choose a subset of exactly p items of the minimum total cost. This problem is a special case of 0-1 knapsack in which all the item weights are equal to 1. Its deterministic version has an O(n)-time algorithm, which consists in choosing p items of the smallest costs. In this paper it is assumed that the item costs are uncertain. Two robust models, namely two-stage and recoverable ones, under discrete and interval uncertainty representations, are discussed. Several positive and negative complexity results for both of them are provided.
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