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 for which the maximum edge congestion is minimized, where the congestion of an edge e of T is the number of vertex pairs adjacent in G for which the path connecting them in T 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 can be efficiently approximated with ratio o(n). In the decision version of this problem, denoted $${K\!-\!\textsf {STC}}$$ , we need to determine if G has a spanning tree with congestion at most K. It is known that $${K\!-\!\textsf {STC}}$$ 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, $${3\!-\!\textsf {STC}}$$ 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 result by proving that $${K\!-\!\textsf {STC}}$$ 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.