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.
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Meer, K. (2011). An Extended Tree-Width Notion for Directed Graphs Related to the Computation of Permanents. In: Kulikov, A., Vereshchagin, N. (eds) Computer Science – Theory and Applications. CSR 2011. Lecture Notes in Computer Science, vol 6651. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20712-9_19
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DOI: https://doi.org/10.1007/978-3-642-20712-9_19
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