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

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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.

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Notes

  1. Note here that if there are several compatible triples involving the same type λ t , then by the compatibility conditions we obtain always the same list of non-active nodes.

  2. Thanks to P. Golovach for pointing to this paper.

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Acknowledgements

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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  1. Lehrstuhl Theoretische Informatik, BTU Cottbus, Platz der Deutschen Einheit 1, 03044, Cottbus, Germany

    Klaus Meer

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  1. Klaus Meer
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Correspondence to Klaus Meer.

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Meer, K. An Extended Tree-Width Notion for Directed Graphs Related to the Computation of Permanents. Theory Comput Syst 55, 330–346 (2014). https://doi.org/10.1007/s00224-013-9490-z

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