Parameterized Spanning Tree Congestion

TL;DR

This paper investigates the computational complexity of the Spanning Tree Congestion problem, establishing its hardness with respect to various graph parameters and degree bounds, and providing new insights into its structural intractability.

Contribution

It proves W[1]-hardness for treewidth and related parameters, NP-hardness on graphs with maximum degree 8, and introduces a generic reduction applicable to multiple parameters.

Findings

Spanning Tree Congestion is W[1]-hard parameterized by treewidth.

NP-hardness holds for graphs with maximum degree 8.

The problem is NP-complete on graphs with modular-width 4.

Abstract

In this paper we study the Spanning Tree Congestion problem, where we are given a graph $G=(V,E)$ and are asked to find a spanning tree $T$ of minimum maximum congestion. Here, the congestion of an edge $e\in T$ is the number of edges $uv\in E$ such that the (unique) path from $u$ to $v$ in $T$ traverses $e$. We consider this well-studied NP-hard problem from the point of view of (structural) parameterized complexity and obtain the following results. We resolve a natural open problem by showing that Spanning Tree Congestion is not FPT parameterized by treewidth (under standard assumptions). More strongly, we present a generic reduction which applies to (almost) any parameter of the form ``vertex-deletion distance to class $\mathcal{C}$'', thus obtaining W[1]-hardness for parameters more restricted than treewidth, including tree-depth plus feedback vertex set, or incomparable to treewidth, such as twin cover. Via a slight tweak of the same reduction we also show that the problem is NP-complete on graphs of modular-width $4$. Even though it is known that Spanning Tree Congestion remains NP-hard on instances with only one vertex of unbounded degree, it is currently open whether the problem remains hard on bounded-degree graphs. We resolve this question by showing NP-hardness on graphs of maximum degree 8. Complementing the problem's W[1]-hardness for treewidth...