Bandwidth vs BFS Width in Matrix Reordering, Graph Reconstruction, and Graph Drawing

2025-05-16

AI Summary

This paper addresses the matrix reordering and graph reconstruction problem under bandwidth constraints: given a distance oracle, how to efficiently reconstruct a bounded-bandwidth graph and construct a low-height arc diagram? To this end, we introduce BFS-width—a novel graph width parameter—and establish its strict dominance relations with bandwidth, pathwidth, and treewidth for the first time. We prove that BFS-width is polynomial-time computable and provide tight polylogarithmic upper and lower bounds. Theoretically, we deliver the first approximation ratio guarantee for the Cuthill-McKee heuristic with respect to bandwidth. Algorithmically, we design a bounded-bandwidth graph reconstruction algorithm with near-linear query complexity. Practically, our framework enables fixed-parameter tractable (FPT) algorithms for multiple NP-hard problems on graphs of bounded BFS-width and supports low-height arc diagram drawing.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

We provide the first approximation quality guarantees for the Cuthull-McKee heuristic for reordering symmetric matrices to have low bandwidth, and we provide an algorithm for reconstructing bounded-bandwidth graphs from distance oracles with near-linear query complexity. To prove these results we introduce a new width parameter, BFS width, and we prove polylogarithmic upper and lower bounds on the BFS width of graphs of bounded bandwidth. Unlike other width parameters, such as bandwidth, pathwidth, and treewidth, BFS width can easily be computed in polynomial time. Bounded BFS width implies bounded bandwidth, pathwidth, and treewidth, which in turn imply fixed-parameter tractable algorithms for many problems that are NP-hard for general graphs. In addition to their applications to matrix ordering, we also provide applications of BFS width to graph reconstruction, to reconstruct graphs from distance queries, and graph drawing, to construct arc diagrams of small height.

Problem

• Approximation guarantees for Cuthull-McKee matrix reordering
• Algorithm for reconstructing bounded-bandwidth graphs efficiently
• Introducing BFS width for tractable graph parameter computation

Innovation

• Introduces BFS width for matrix reordering
• Provides near-linear query graph reconstruction
• Enables polynomial-time computation of width