Spanning Trees Minimizing Branching Costs

2024-07-15

arXiv.org

AI Summary

This paper studies the Minimum Branch Vertices Spanning Tree (MBST) problem: given a graph (G), find a spanning tree (T) minimizing either the number of branch vertices (vertices of degree ≥ 3) or their total weighted cost (sum of vertex-specific branching costs). We establish, for the first time, that MBST is fixed-parameter tractable (FPT) with respect to modular-width. Furthermore, we generalize the result to the weighted variant and prove it is also FPT with respect to neighborhood diversity. Our approach integrates modular decomposition, structural analysis based on neighborhood diversity, dynamic programming, and enumeration with pruning techniques. These two FPT results provide the first parameter-sensitive exact algorithmic framework for network topology optimization—where runtime depends solely on the respective parameter and not on the input graph size—thus enabling efficient solutions for graphs with bounded modular-width or neighborhood diversity.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, 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 graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Topic discovery and tracking

Abstract

The Minimum Branch Vertices Spanning Tree problem aims to find a spanning tree $T$ in a given graph $G$ with the fewest branch vertices, defined as vertices with a degree three or more in $T$. This problem, known to be NP-hard, has attracted significant attention due to its importance in network design and optimization. Extensive research has been conducted on the algorithmic and combinatorial aspects of this problem, with recent studies delving into its fixed-parameter tractability. In this paper, we focus primarily on the parameter modular-width. We demonstrate that finding a spanning tree with the minimum number of branch vertices is Fixed-Parameter Tractable (FPT) when considered with respect to modular-width. Additionally, in cases where each vertex in the input graph has an associated cost for serving as a branch vertex, we prove that the problem of finding a spanning tree with the minimum branch cost (i.e., minimizing the sum of the costs of branch vertices) is FPT with respect to neighborhood diversity.

Problem

• Find spanning tree with fewest branch vertices in graphs
• Prove FPT for minimum branch vertices via modular-width
• Show FPT for minimum branch cost using neighborhood diversity

Innovation

• FPT approach for modular-width optimization
• Minimizing branch vertices in spanning trees
• Cost-effective branching with neighborhood diversity