Bandwidth and Distortion Revisited

TL;DR

This paper introduces faster exact algorithms for the bandwidth and distortion problems in graphs, improving previous methods with algorithms that operate in exponential time but polynomial space, advancing the computational efficiency of these graph embedding problems.

Contribution

The paper develops new algorithms for bandwidth and distortion problems with improved exponential time bounds and polynomial space complexity, including the first polynomial space algorithm for distortion.

Findings

Algorithms run in O(9.363^n) time for both problems.

Improves previous algorithms for bandwidth and distortion.

Provides the first polynomial space exact algorithm for distortion.

Abstract

In this paper we merge recent developments on exact algorithms for finding an ordering of vertices of a given graph that minimizes bandwidth (the BANDWIDTH problem) and for finding an embedding of a given graph into a line that minimizes distortion (the DISTORTION problem). For both problems we develop algorithms that work in O(9.363^n) time and polynomial space. For BANDWIDTH, this improves O^*(10^n) algorithm by Feige and Kilian from 2000, for DISTORTION this is the first polynomial space exact algorithm that works in O(c^n) time we are aware of. As a byproduct, we enhance the O(5^{n+o(n)})-time and O^*(2^n)-space algorithm for DISTORTION by Fomin et al. to an algorithm working in O(4.383^n) time and space.