Theory of Computing Systems

, Volume 55, Issue 2, pp 330–346

# An Extended Tree-Width Notion for Directed Graphs Related to the Computation of Permanents

Article

## 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 for directed graphs 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 is shown to hold as well for the Hamiltonian Cycle problem.

## Keywords

Treewidth notions for digraphs Efficient algorithms Computation of the permanent

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