An Exponential Time 2-Approximation Algorithm for Bandwidth

The bandwidth of a graph G on n vertices is the minimum b such that the vertices of G can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b. In this paper, we present a 2-approximation algorithm for the Bandwidth problem that takes worst-case $\mathcal{O}(1.9797^n)$ $= \mathcal{O}(3^{0.6217 n})$ time and uses polynomial space. This improves both the previous best 2- and 3-approximation algorithms of Cygan et al. which have an $\mathcal{O}^*(3^n)$ and $\mathcal{O}^*(2^n)$ worst-case time bounds, respectively. Our algorithm is based on constructing bucket decompositions of the input graph. A bucket decomposition partitions the vertex set of a graph into ordered sets (called buckets) of (almost) equal sizes such that all edges are either incident on vertices in the same bucket or on vertices in two consecutive buckets. The idea is to find the smallest bucket size for which there exists a bucket decomposition. The algorithm uses a simple divide-and-conquer strategy along with dynamic programming to achieve this improved time bound.