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.KeywordsFast Fourier TransformApproximation AlgorithmChromatic NumberExact AlgorithmInput GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.