Improved Lower Bounds for the Cyclic Bandwidth Problem

Abstract

We study the classical Cyclic Bandwidth problem, an optimization problem which takes as input an undirected graph $G=(V,E)$ with $|V|=n$, and asks for a labeling $\varphi$ of V in which every vertex v takes a unique value $\varphi (v)\in [1;n]$, in such a way that $B_c(G,\varphi )=\max \{\min _{uv\in E(G)}\{|\varphi (u)-\varphi (v)|,n-|\varphi (u)-\varphi (v)|\}\}$, called the cyclic bandwidth of G, is minimized.

We provide three new and improved lower bounds for the Cyclic Bandwidth problem, applicable to any graph G: two are based on the neighborhood vertex density of G, the other one on the length of a longest cycle in a cycle basis of G. We also show that our results improve the best known lower bounds for a large proportion of a set of instances taken from a frequently used benchmark, the Harwell-Boeing sparse matrix collection. Our third proof provides additional elements: first, an improved sufficient condition yielding $B_c(G)=B(G)$ (where $B(G)=\min _{\varphi }\{\max _{uv\in E(G)}\{|\varphi (u)-\varphi (v)|\}\}$ denotes the bandwidth of G) ; second, an algorithm that, under some conditions, computes a labeling reaching B(G) from a labeling reaching $B_c(G)$.