On the Complexity of the Bilevel Minimum Spanning Tree Problem

TL;DR

This paper investigates the computational complexity of the bilevel minimum spanning tree problem, proving NP-hardness in general and special cases, and introduces approximation algorithms and fixed-parameter tractability results.

Contribution

It establishes NP-hardness for BMST, provides approximation algorithms, and explores fixed-parameter tractability for various problem variants.

Findings

BMST is NP-hard in general and even when the follower controls only a matching.

A polynomial-time $(|V|-1)$-approximation algorithm for BMST.

BMST is fixed-parameter tractable when parameterized by the number of follower-controlled edges.

Abstract

We consider the bilevel minimum spanning tree (BMST) problem where the leader and the follower choose a spanning tree together, according to different objective functions. By showing that this problem is NP-hard in general, we answer an open question stated in by Shi et al. We prove that BMST remains hard even in the special case where the follower only controls a matching. Moreover, by a polynomial reduction from the vertex-disjoint Steiner trees problem, we give some evidence that BMST might even remain hard in case the follower controls only few edges. On the positive side, we present a polynomial-time $(|V|-1)$-approximation algorithm for BMST, where $|V|$ is the number of vertices in the input graph. Moreover, considering the number of edges controlled by the follower as parameter, we show that 2-approximating BMST is fixed-parameter tractable and that, in case of uniform costs on leader's edges, even solving BMST exactly is fixed-parameter tractable. We finally consider bottleneck variants of BMST and settle the complexity landscape of all combinations of sum or bottleneck objective functions for the leader and follower, for the optimistic as well as the pessimistic setting.