Making Robust Decisions in Discrete Optimization Problems as a Game against Nature

In this paper a discrete optimization problem under uncertainty is discussed. Solving such a problem can be seen as a game against nature. In order to choose a solution, the minmax and minmax regret criteria can be applied. In this paper an extension of the known minmax (regret) approach is proposed. It is shown how different types of uncertainty can be simultaneously taken into account. Some exact and approximation algorithms for choosing a best solution are constructed.Discrete optimization, minmax, minmax regret, game against nature

Optimal Multispecies Harvesting in the Presence of a Nuisance Species

Crop Production/Industries,

On the approximability of robust spanning tree problems

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 $O(\log^{1-\epsilon} n)$ for any $\epsilon>0$ 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

Complexity of the robust weighted independent set problems on interval graphs

This paper deals with the max-min and min-max regret versions of the maximum weighted independent set problem on interval graphswith uncertain vertex weights. Both problems have been recently investigated by Nobibon and Leus (2014), who showed that they are NP-hard for two scenarios and strongly NP-hard if the number of scenarios is a part of the input. In this paper, new complexity and approximation results on the problems under consideration are provided, which extend the ones previously obtained. Namely, for the discrete scenario uncertainty representation it is proven that if the number of scenarios $K$ is a part of the input, then the max-min version of the problem is not at all approximable. On the other hand, its min-max regret version is approximable within $K$ and not approximable within $O(\log^{1-\epsilon}K)$ for any $\epsilon>0$ unless the problems in NP have quasi polynomial algorithms. Furthermore, for the interval uncertainty representation it is shown that the min-max regret version is NP-hard and approximable within 2