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Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

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Abstract

We consider the multivariate interlace polynomial introduced by Courcelle (Electron. J. Comb. 15(1), 2008), which generalizes several interlace polynomials defined by Arratia, Bollobás, and Sorkin (J. Comb. Theory Ser. B 92(2):199–233, 2004) and by Aigner and van der Holst (Linear Algebra Appl., 2004). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle, Electron. J. Comb. 15(1), 2008) employs a general logical framework and leads to an algorithm with running time f(k)⋅n, where f(k) is doubly exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses \(2^{3k^{2}+O(k)}\cdot n\) arithmetic operations and can be efficiently implemented in parallel.

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Correspondence to Christian Hoffmann.

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Bläser, M., Hoffmann, C. Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth. Algorithmica 61, 3–35 (2011). https://doi.org/10.1007/s00453-010-9439-4

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Keywords

  • Parameterized algorithm
  • Tree decomposition
  • Interlace polynomial
  • Adjacency matrix
  • Gaussian elimination