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 an O ( n O ( t ) 2 n ) exact algorithm. Moreover, for any two positive integers k ≥ 2 , r ≥ 1 , we present a ( 2 k r − 1 ) -approximation algorithm that solves Bandwidth for an arbitrary input graph in O ∗ ( k n ( k − 1 ) r ) time and polynomial space where by O ∗ we denote the standard big O notation but omitting polynomial factors. Finally, we improve the currently best known exact algorithm for arbitrary graphs with an O ( 4.38 3 n ) 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 ∗ ( 2 n ) time and space, what matches the best known results.