# Lower Bounds for the Bandwidth Problem

## TL;DR

This paper introduces a new method using vertex partitions and semidefinite programming to derive lower bounds for the graph bandwidth problem, improving the trade-off between bound quality and computational efficiency.

## Contribution

It presents a novel approach that generalizes previous bounds and incorporates SDP relaxation for better lower bounds on graph bandwidth.

## Key findings

- The approach generalizes existing bounds.
- Semidefinite programming improves bound accuracy.
- Effective on real-world graph instances.

## Abstract

The Bandwidth Problem seeks for a simultaneous permutation of the rows and columns of the adjacency matrix of a graph such that all nonzero entries are as close as possible to the main diagonal. This work focuses on investigating novel approaches to obtain lower bounds for the bandwidth problem. In particular, we use vertex partitions to bound the bandwidth of a graph. Our approach contains prior approaches for bounding the bandwidth as special cases. By varying sizes of partitions, we achieve a trade-off between quality of bounds and efficiency of computing them. To compute lower bounds, we derive a Semidefinite Programming relaxation. We evaluate the performance of our approach on several data sets, including real-world instances.