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

TL;DR

This paper introduces BFS width, a new graph parameter with polynomial-time computability, providing approximation guarantees for matrix reordering, graph reconstruction, and drawing, and establishing bounds relating BFS width to other parameters.

Contribution

The paper defines BFS width, proves bounds relating it to other parameters, and applies it to matrix reordering, graph reconstruction, and graph drawing problems.

Findings

BFS width can be computed in polynomial time.

Bounds relate BFS width to bandwidth, pathwidth, and treewidth.

Applications include improved algorithms for matrix reordering and graph visualization.

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.