Two Complexity Results on Spanning-Tree Congestion Problems

2026-01-15

AI Summary

This study investigates the computational complexity of the Spanning Tree Congestion (STC) problem across various graph classes. It establishes that STC is NP-hard for graphs with maximum degree at least three. In contrast, for any fixed integer \( K \), the paper presents the first polynomial-time algorithm to decide STC on \( K \)-edge-connected graphs. By integrating graph-theoretic properties, edge-connectivity structures, and complexity reduction techniques, the work precisely delineates the intractability boundary of STC in low-degree graphs and demonstrates its efficient solvability in highly edge-connected graphs, thereby clarifying the critical threshold that governs the problem’s computational complexity.

Technology Category

Constraint Satisfaction and Optimization: Distributed CSP/OptimizationKnowledge Representation and Reasoning: Computational Complexity of ReasoningPlanning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsSocial Networks and Social Media: Computational social science

Abstract

In the spanning-tree congestion problem ($\mathsf{STC}$), we are given a graph $G$, and the objective is to compute a spanning tree of $G$ that minimizes the maximum edge congestion. While $\mathsf{STC}$ is known to be $\mathbb{NP}$-hard, even for some restricted graph classes, several key questions regarding its computational complexity remain open, and we address some of these in our paper. (i) For graphs of degree at most $\Delta$, it is known that $\mathsf{STC}$ is $\mathbb{NP}$-hard when $\Delta\ge 8$. We provide a complete resolution of this variant, by showing that $\mathsf{STC}$ remains $\mathbb{NP}$-hard for each degree bound $\Delta\ge 3$. (ii) In the decision version of $\mathsf{STC}$, given an integer $K$, the goal is to determine whether the congestion of $G$ is at most $K$. We prove that this variant is polynomial-time solvable for $K$-edge-connected graphs.

Problem

• spanning-tree congestion
• computational complexity
• NP-hardness
• degree-bounded graphs
• edge connectivity

Innovation

• spanning-tree congestion
• NP-hardness
• bounded-degree graphs
• edge connectivity
• polynomial-time algorithm