Theory of Computing Systems

, Volume 55, Issue 2, pp 330–346 | Cite as

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

  • Klaus MeerEmail author


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.


Treewidth notions for digraphs Efficient algorithms Computation of the permanent 



Thanks are due to the anonymous referees for helpful comments improving readability of the paper and for pointing out some open problems.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Lehrstuhl Theoretische InformatikBTU CottbusCottbusGermany

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