Approximating Spanning Tree Congestion on Graphs with Polylog Degree

Given a graph G and a spanning tree T of G, the congestion of an edge \(e\in E(T)\) , with respect to G and T, is the number of edges uv in G such that the unique path in T connecting the vertices u and v traverses the edge e. Given a connected graph G, the spanning tree congestion problem is to construct a spanning tree T that minimizes its maximum edge congestion. It is known that the problem is NP-hard, and that every spanning tree is an n/2-approximation, but it is not even known whether an o(n)-approximation is possible in polynomial time; by n we denote the number of vertices in the graph G. We consider the problem on graphs with maximum degree bounded by \(\varDelta =polylog(n)\) and describe an o(n)-approximation algorithm; note that even on this restricted class of graphs the spanning tree congestion can be of order \(n\cdot polylog(n)\) .