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Algorithmica

, Volume 73, Issue 4, pp 696–729 | Cite as

Myhill–Nerode Methods for Hypergraphs

  • René van Bevern
  • Rodney G. Downey
  • Michael R. Fellows
  • Serge Gaspers
  • Frances A. Rosamond
Article

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.

Keywords

NP-hard problems Fixed-parameter algorithms Automata theory Cutwidth Hypertree width 

Notes

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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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • René van Bevern
    • 1
  • Rodney G. Downey
    • 2
  • Michael R. Fellows
    • 3
  • Serge Gaspers
    • 4
  • Frances A. Rosamond
    • 3
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany
  2. 2.Victoria University of WellingtonWellingtonNew Zealand
  3. 3.School of Engineering and ITCharles Darwin UniversityDarwinAustralia
  4. 4.The University of New South Wales and NICTASydneyAustralia

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