Better Hardness Results for the Minimum Spanning Tree Congestion Problem

Abstract

In the spanning tree congestion problem, given a connected graph G, the objective is to compute a spanning tree T in G that minimizes its maximum edge congestion, where the congestion of an edge e of T is the number of edges in G for which the unique path in T between their endpoints traverses e. The problem is known to be \(\mathbb{N}\mathbb{P}\)-hard, but its approximability is still poorly understood, and it is not even known whether the optimum solution can be efficiently approximated with ratio o(n). In the decision version of this problem, denoted \({\varvec{K}-\textsf {STC}}\), we need to determine if G has a spanning tree with congestion at most K. It is known that \({\varvec{K}-\textsf {STC}}\) is \(\mathbb{N}\mathbb{P}\)-complete for \(K\ge 8\), and this implies a lower bound of 1.125 on the approximation ratio of minimizing congestion. On the other hand, \({\varvec{3}-\textsf {STC}}\) can be solved in polynomial time, with the complexity status of this problem for \(K\in { \left\{ 4,5,6,7 \right\} }\) remaining an open problem. We substantially improve the earlier hardness results by proving that \({\varvec{K}-\textsf {STC}}\) is \(\mathbb{N}\mathbb{P}\)-complete for \(K\ge 5\). This leaves only the case \(K=4\) open, and improves the lower bound on the approximation ratio to 1.2. Motivated by evidence that minimizing congestion is hard even for graphs of small constant radius, we also consider \({\varvec{K}-\textsf {STC}}\) restricted to graphs of radius 2, and we prove that this variant is \(\mathbb{N}\mathbb{P}\)-complete for all \(K\ge 6\).