Complexity Results for Bandwidth Minimization

Abstract

We present a linear-time algorithm for sparse symmetric matrices which converts a matrix into pentadiagonal form (“bandwidth 2”), whenever it is possible to do so using simultaneous row and column permutations. On the other hand when an arbitrary integer k and graph G are given, we show that it is $NP$-complete to determine whether or not there exists an ordering of the vertices such that the adjacency matrix has bandwidth $ \leqq k$, even when G is restircted to the class of free trees with all vertices of degree $ \leqq 3$. Related problems for acyclic directed graphs (upper triangular matrices) are also discussed.