Parameterized Spanning Tree Congestion

2024-10-10

International Symposium on Mathematical Foundations of Computer Science

AI Summary

This paper studies the Tree Congestion Minimization problem: given a graph (G = (V,E)), compute a spanning tree (T) minimizing the maximum number of vertex-pair unique paths in (T) traversing any single edge—i.e., the edge congestion. While known to be NP-hard, its parameterized complexity remained open for years. We resolve this by proving, under the Exponential Time Hypothesis (ETH), that the problem is not fixed-parameter tractable (FPT) with respect to treewidth. Using a novel generic reduction framework, we establish W[1]-hardness with respect to stronger or incomparable structural parameters—including tree-depth plus feedback vertex set, and twin cover. Furthermore, we show NP-completeness even on graphs with maximum degree (Delta = 8) and modular width (mathrm{mw} = 4). These results comprehensively settle multiple long-standing open questions and significantly advance the theoretical boundaries of structural parameterized algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $ein T$ is the number of edges $uvin E$ 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...

Problem

• Resolving FPT status for treewidth parameterization in congestion
• Establishing W[1]-hardness for restricted structural parameters
• Proving NP-hardness on bounded-degree graphs with maximum degree 8

Innovation

• Proved W[1]-hardness for treewidth parameterization
• Showed NP-hardness on graphs with maximum degree 8
• Established NP-completeness for modular-width 4 graphs