Approximation of Spanning Tree Congestion using Hereditary Bisection

TL;DR

The Spanning Tree Congestion (STC) problem seeks a spanning tree that minimizes the maximum edge congestion, a notoriously hard challenge with only the trivial $n/2$-approximation previously available. The authors introduce a new lower bound $STC(G)\ge \Omega\left(\frac{hb(G)}{\Delta}\right)$ linking STC to hereditary bisection width $hb(G)$, and couple it with a divide-and-conquer algorithm (CongSpanTree) that achieves an $O(\Delta\cdot \log^{3/2} n)$-approximation in general graphs and $O(\Delta\log n)$ for minor-free graphs; an ARV/KleinPR-based cut-approximation factor $\alpha(n)$ governs the recursion. The approach yields sublinear guarantees for broad graph classes (e.g., graphs with modest maximum degree) and broadens the toolkit by tying STC to hereditary properties of subgraphs, enabling improved analyses and potential applications. The results thus significantly strengthen the prospects for sublinear STC approximations and open questions about removing degree依ndence and extending hereditary-bisection techniques to other combinatorial problems.

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 $O(Δ\cdot\log^{3/2}n)$-approximation algorithm where $Δ$ 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 $G$, $STC(G)\geq Ω(hb(G)/Δ)$.

Theorems & Definitions (8)

• Theorem 1.1: ARV:09KleinPR:93
• Theorem 1.2: Jordan Jordan:69
• Lemma 2.1
• Corollary 2.2
• Lemma 2.3
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3