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Myhill–Nerode Methods for Hypergraphs

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Abstract

We give an analog of the Myhill–Nerode theorem from formal language theory for hypergraphs and use it to derive the following results for two NP-hard hypergraph problems. (1) We provide an algorithm for testing whether a hypergraph has cutwidth at most \(k\) that runs in linear time for constant \(k\). In terms of parameterized complexity theory, the problem is fixed-parameter linear parameterized by \(k\). (2) We show that it is not expressible in monadic second-order logic whether a hypergraph has bounded (fractional, generalized) hypertree width. The proof leads us to conjecture that, in terms of parameterized complexity theory, these problems are W[1]-hard parameterized by the incidence treewidth (the treewidth of the incidence graph). Thus, in the form of the Myhill–Nerode theorem for hypergraphs, we obtain a method to derive linear-time algorithms and to obtain indicators for intractability for hypergraph problems parameterized by incidence treewidth.

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Notes

  1. For keeping the presentation simpler, we leave graphs with less than \(t\) vertices out of consideration: since we only consider constant values of \(t\) throughout this work, any problem restricted to graphs with less than \(t\) vertices is a finite problem and can therefore be trivially solved by a finite automaton.

  2. The proof of (ii)\({}\rightarrow {}\)(i) given by Downey and Fellows [17] is flawed but repairable [4].

  3. If \(F\) is not generator-total, it might be that \(G\notin F\) but \(\mathcal H(G)\in \mathcal H(F)\) because \(H\in F\) for some \(t\)-boundaried hypergraph generator \(H\ne G\) with \(\mathcal H(G)=\mathcal H(H)\): the graphs \(G\) and \(H\) might represent the hyperedges of \(\mathcal H(G)=\mathcal H(H)\) using different mathematical objects.

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Acknowledgments

The authors are thankful to Mahdi Parsa for fruitful discussions. René van Bevern acknowledges support by the Deutsche Forschungsgesellschaft (DFG), Project DAPA (NI 369/12). Rod Downey acknowleges support by a grant from the New Zealand Marsden Fund. The remaining three authors acknowledge support by the Australian Research Council, Grants DP 1097129 (Michael R. Fellows), DE 120101761 (Serge Gaspers), and DP 110101792 (Michael R. Fellows and Frances A. Rosamond). NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.

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Authors and Affiliations

  1. Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Berlin, Germany

    René van Bevern

  2. Victoria University of Wellington, Wellington, New Zealand

    Rodney G. Downey

  3. School of Engineering and IT, Charles Darwin University, Darwin, Australia

    Michael R. Fellows & Frances A. Rosamond

  4. The University of New South Wales and NICTA, Sydney, Australia

    Serge Gaspers

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  1. René van Bevern
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  2. Rodney G. Downey
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  3. Michael R. Fellows
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  4. Serge Gaspers
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  5. Frances A. Rosamond
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Corresponding author

Correspondence to René van Bevern.

Additional information

A preliminary version of this article appeared in the proceedings of ISAAC 2013 [5]. This extended and revised version contains the full proof details, more figures, and corollaries to make the application of the Myhill–Nerode theorem for hypergraphs easier in an algorithmic setting.

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van Bevern, R., Downey, R.G., Fellows, M.R. et al. Myhill–Nerode Methods for Hypergraphs. Algorithmica 73, 696–729 (2015). https://doi.org/10.1007/s00453-015-9977-x

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