Approximation of Spanning Tree Congestion using Hereditary Bisection

TL;DR

This paper presents a new approximation algorithm for the NP-hard Spanning Tree Congestion problem, achieving significant improvements for graphs with bounded maximum degree, and introduces a novel lower bound based on hereditary bisection.

Contribution

The authors develop an $O(\Delta\cdot\log^{3/2}n)$-approximation algorithm for STC and establish a new lower bound involving hereditary bisection width.

Findings

The algorithm provides exponential improvement for graphs with polylogarithmic maximum degree.

A new lower bound on STC is proven using hereditary bisection width.

The approach links graph bisection properties to spanning tree congestion.

Abstract

The Spanning Tree Congestion (STC) problem is the following NP-hard problem: given a graph $G$, construct a spanning tree $T$ of $G$ minimizing its maximum edge congestion where the congestion of an edge $e\in T$ is the number of edges $uv$ in $G$ such that the unique path between $u$ and $v$ in $T$ passes through $e$; the optimal value for a given graph $G$ is denoted $STC(G)$. It is known that every spanning tree is an $n/2$-approximation for the STP problem. A long-standing problem is to design a better approximation algorithm. Our contribution towards this goal is an $O(\Delta\cdot\log^{3/2}n)$-approximation algorithm where $\Delta$ is the maximum degree in $G$ and $n$ the number of vertices. For graphs with a maximum degree bounded by a polylog of the number of vertices, this is an exponential improvement over the previous best approximation. Our main tool for the algorithm is a new lower bound on the spanning tree congestion which is of independent interest. Denoting by $hb(G)$ the hereditary bisection of $G$ which is the maximum bisection width over all subgraphs of $G$, we prove that for every graph $G$, $STC(G)\geq \Omega(hb(G)/\Delta)$.