Exact and Approximate Bandwidth

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(n2) 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 + √ n/α) 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 r ) time and polynomial space. Finally we improve the currently best known exact algorithm for arbitrary graphs with a O(4.473) 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.