Approximability of the robust representatives selection problem

TL;DR

This paper investigates the computational complexity and approximation limits of the robust representatives selection problem under scenario uncertainty, establishing inapproximability bounds and providing an approximation algorithm.

Contribution

It extends previous results by proving new inapproximability bounds and offering an approximation algorithm for the problem with multiple scenarios.

Findings

Min-max representatives selection problem is not approximable within $O(\log^{1-\epsilon}K)$ unless NP problems have quasi-polynomial algorithms.

An $O(rac{\log K}{\log \log K})$ approximation algorithm is proposed for the min-max version.

The results generalize and improve upon earlier complexity and approximation bounds.

Abstract

In this paper new complexity and approximation results on the robust versions of the representatives selection problem, under the scenario uncertainty representation, are provided, which extend the results obtained in the recent papers by Dolgui and Kovalev (2012), and Deineko and Woeginger (2013). Namely, it is shown that if the number of scenarios is a part of input, then the min-max (regret) representatives selection problem is not approximable within a ratio of $O(\log^{1-\epsilon}K)$ for any $\epsilon>0$, where $K$ is the number of scenarios, unless the problems in NP have quasi-polynomial time algorithms. An approximation algorithm with an approximation ratio of $O(\log K/ \log \log K)$ for the min-max version of the problem is also provided.