Hardness Results and an Exact Exponential Algorithm for the Spanning Tree Congestion Problem

Abstract

Spanning tree congestion is a relatively new graph parameter, which has been studied intensively. This paper studies the complexity of the problem to determine the spanning tree congestion for non-sparse graph classes, while it was investigated for some sparse graph classes before. We prove that the problem is NP-hard even for chain graphs and split graphs. To cope with the hardness of the problem, we present a fast (exponential-time) exact algorithm that runs in O^∗(2ⁿ) time, where n denotes the number of vertices. Additionally, we present simple combinatorial lemmas, which yield a constant-factor approximation algorithm for cographs, and a linear-time algorithm for chordal cographs.