Graph Bandwidth

Keywords and Synonyms

Graph bandwidth; Approximation algorithms; Metric embeddings      

Problem Definition

The graph bandwidth problem concerns producing a linear ordering of the vertices of a graph ${ G = (V,E) }$ so as to minimize the maximum “stretch” of any edge in the ordering. Formally, let ${ n = |V| }$, and consider any one-to-one mapping ${ \pi : V \to \{1,2, \dots, n\} }$. The bandwidth of this ordering is ${ \mathsf{bw}_\pi(G) = \max_{\{u,v\} \in E} |\pi(u)-\pi(v)| }$. The bandwidth of G is given by the bandwidth of the best possible ordering: ${ \mathsf{bw}(G) = \min_{\pi} \mathsf{bw}_\pi(G) }$.

The original motivation for this problem lies in the preprocessing of sparse symmetric square matrices. Let A be such an ${ n \times n }$ matrix, and consider the problem of finding a permutation matrix P such that the non-zero entries of ${ P^{\text{T}} A P }$all lie in as narrow a band as possible about the diagonal. This problem is equivalent to...