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 $\mathbb{K}$ 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.