Complexity of the robust weighted independent set problems on interval graphs

TL;DR

This paper investigates the computational complexity of robust weighted independent set problems on interval graphs under uncertainty, providing new hardness and approximation results for different scenario models.

Contribution

It extends previous work by establishing new complexity and approximation bounds for max-min and min-max regret versions under discrete and interval uncertainty.

Findings

Max-min problem not approximable when scenarios are part of input.

Min-max regret problem approximable within factor K.

Interval uncertainty min-max regret problem is NP-hard and 2-approximable.

Abstract

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.