On the approximability of robust spanning tree problems

TL;DR

This paper investigates the computational difficulty of robust spanning tree problems under uncertain edge costs, establishing hardness results and proposing approximation algorithms with performance guarantees.

Contribution

It analyzes the approximability of min-max, min-max regret, and 2-stage min-max spanning tree problems, providing new hardness bounds and LP-based approximation algorithms.

Findings

Min-max and min-max regret problems are hard to approximate within $O(\log^{1-\epsilon} n)$.

2-stage min-max problem cannot be approximated within $O(\log n)$ unless NP problems have quasi-polynomial algorithms.

Proposed randomized LP-based algorithms achieve an $O(\log^2 n)$ approximation ratio.

Abstract

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.