Exact and Approximate Bandwidth

Abstract

In this paper we gather several improvements in the field of exact and approximate exponential-time algorithms for the Bandwidth problem. For graphs with treewidth t we present a O(n ^O(t) 2ⁿ) exact algorithm. Moreover for the same class of graphs we introduce a subexponential constant-approximation scheme – for any α> 0 there exists a (1 + α)-approximation algorithm running in $O(\exp(c(t + \sqrt{n/\alpha})\log n))$ time where c is a universal constant. These results seem interesting since Unger has proved that Bandwidth does not belong to APX even when the input graph is a tree (assuming P ≠ NP). So somewhat surprisingly, despite Unger’s result it turns out that not only a subexponential constant approximation is possible but also a subexponential approximation scheme exists. Furthermore, for any positive integer r, we present a (4r − 1)-approximation algorithm that solves Bandwidth for an arbitrary input graph in $O^*(2^{n\over r})$ time and polynomial space. Finally we improve the currently best known exact algorithm for arbitrary graphs with a O(4.473ⁿ) time and space algorithm.

In the algorithms for the small treewidth we develop a technique based on the Fast Fourier Transform, parallel to the Fast Subset Convolution techniques introduced by Björklund et al. This technique can be also used as a simple method of finding a chromatic number of all subgraphs of a given graph in O ^*(2ⁿ) time and space, what matches the best known results.