An extended tree-width notion for directed graphs related to the computation of permanents

Abstract

It is well known that permanents of matrices of bounded tree-width are efficiently computable. Here, the tree-width of a square matrix M = (m ij ) with entries from a field is the tree-width of the underlying graph G M having an edge (i,j) if and only if the entry m ij 0. Though G M is directed this does not influence the tree-width definition. Thus, it does not reflect the lacking symmetry when m ij 0 but m ji = 0. The latter however might have impact on the computation of the permanent. In this paper we introduce and study an extended notion of tree-width called triangular tree-width. We give examples where the latter parameter is bounded whereas the former is not. As main result we show that permanents of matrices of bounded triangular tree-width are efficiently computable. This result holds as well for the Hamiltonian Cycle problem. © 2011 Springer-Verlag Berlin Heidelberg.