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

Myhill–Nerode Methods for Hypergraphs

  • René van BevernEmail author
  • Rodney G. Downey
  • Michael R. Fellows
  • Serge Gaspers
  • Frances A. Rosamond


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.


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



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.


  1. 1.
    Abrahamson, K.R., Fellows, M.R.: Cutset Regularity Beats Well-Quasi-Ordering for Bounded Treewidth. Tech. rep., Dept. Computer Science, University Victoria, Canada (1989)Google Scholar
  2. 2.
    Abrahamson, K.R., Fellows, M.R.: Finite automata, bounded treewidth, and well-quasiordering. In: Graph Structure Theory, American Mathematical Society, Contemporary Mathematics, vol. 147, pp. 539–564 (1991)Google Scholar
  3. 3.
    Bern, M.W., Lawler, E.L., Wong, A.L.: Why certain subgraph computations require only linear time. In: Proceedings of the 26th FOCS, IEEE Computer Society, pp. 117–125 (1985)Google Scholar
  4. 4.
    van Bevern, R., Downey, R.G., Fellows, M.R., Gaspers, S., Rosamond, F.A.: Myhill–Nerode Methods for Hypergraphs. arXiv:1211.1299v5 [cs.DM] (2015)
  5. 5.
    van Bevern, R., Fellows, M.R., Gaspers, S., Rosamond, F.A.: Myhill–Nerode methods for hypergraphs. In: Proceedings of the 24th ISAAC, LNCS, vol. 8283, pp. 372–382. Springer, Berlin (2013)Google Scholar
  6. 6.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21(2), 358–402 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Bodlaender, H.L., Fellows, M.R., Warnow, T.J.: Two strikes against perfect phylogeny. In: Proceedings of the 19th ICALP, LNCS, vol. 623, pp. 273–283. Springer, Berlin (1992)Google Scholar
  9. 9.
    Bodlaender, H.L., Fellows, M.R., Hallett, M.T.: Beyond NP-completeness for problems of bounded width (extended abstract): hardness for the W hierarchy. In: Proceedings of the 26th STOC, pp. 449–458. ACM (1994)Google Scholar
  10. 10.
    Bodlaender, H.L., Fellows, M.R., Hallett, M.T., Wareham, H.T., Warnow, T.J.: The hardness of perfect phylogeny, feasible register assignment and other problems on thin colored graphs. Theor. Comput. Sci. 244(1–2), 167–188 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Bodlaender, H.L., Fellows, M.R., Thilikos, D.M.: Derivation of algorithms for cutwidth and related graph layout parameters. J. Comput. Syst. Sci. 75(4), 231–244 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Borie, R.B., Parker, R.G., Tovey, C.A.: Solving problems on recursively constructed graphs. ACM Comput. Surv. 41(1) (2009). doi: 10.1145/1456650.1456654
  13. 13.
    Cahoon, J., Sahni, S.: Exact algorithms for special cases of the board permutation problem. In: Proceedings of the 21st Annual Allerton Conference on Communication, Control, and Computing, pp. 246–255 (1983)Google Scholar
  14. 14.
    Courcelle, B., Engelfriet, J.: Graph Structure and Monadic Second-Order Logic—A Language-Theoretic Approach, Encyclopedia of mathematics and Its Applications, vol. 138. Cambridge University Press, Cambridge (2012)Google Scholar
  15. 15.
    Courcelle, B., Lagergren, J.: Equivalent definitions of recognizability for sets of graphs of bounded tree-width. Math. Struct. Comput. Sci. 6(2), 141–165 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)CrossRefGoogle Scholar
  17. 17.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    Fellows, M., Langston, M.: An analogue of the Myhill–Nerode theorem and its use in computing finite-basis characterizations. In: Proceedings of the 30th FOCS, pp. 520–525. IEEE Computer Society (1989)Google Scholar
  19. 19.
    Fellows, M.R., Langston, M.A.: On well-partial-order theory and its application to combinatorial problems of VLSI design. SIAM J. Discrete Math. 5(1), 117–126 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Fellows, M.R., Langston, M.A.: On search, decision, and the efficiency of polynomial-time algorithms. J. Comput. Syst. Sci. 49(3), 769–779 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Fellows, M.R., Jansen, B.M.P., Rosamond, F.: Towards fully multivariate algorithmics: parameter ecology and the deconstruction of computational complexity. Eur. J. Combin. 34(3), 541–566 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)Google Scholar
  23. 23.
    Fomin, F.V., Golovach, P.A., Thilikos, D.M.: Approximating acyclicity parameters of sparse hypergraphs. In: Proceedings of the 26th STACS, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, LIPIcs, vol. 3, pp. 445–456 (2009)Google Scholar
  24. 24.
    Ganian, R., Hliněný, P.: On parse trees and Myhill–Nerode-type tools for handling graphs of bounded rank-width. Discrete Appl. Math. 158(7), 851–867 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Garey, M.R., Graham, R.L., Johnson, D.S., Knuth, D.E.: Complexity results for bandwidth minimization. SIAM J. Appl. Math. 34(3), 477–495 (1978)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Gaspers, S., Naroditskiy, V., Narodytska, N., Walsh, T.: Possible and necessary winner problem in social polls. In: Proceedings of the AAMAS’13, IFAAMAS, pp. 1131–1132 (2013)Google Scholar
  27. 27.
    Gavril, F.: Some NP-complete problems on graphs. In: Proceedings of the 1977 Conference on Information Science and Systems, Johns Hopkins University, pp. 91–95 (1977)Google Scholar
  28. 28.
    Gottlob, G., Grohe, M., Musliu, N., Samer, M., Scarcello, F.: Hypertree decompositions: structure, algorithms, and applications. In: Proceedings of the 31st WG, LNCS, vol. 3787, pp. 1–15. Springer, Berlin (2005)Google Scholar
  29. 29.
    Gottlob, G., Miklós, Z., Schwentick, T.: Generalized hypertree decompositions: NP-hardness and tractable variants. J. ACM 56(6) (2009). doi: 10.1145/1568318.1568320
  30. 30.
    Hliněný, P.: Branch-width, parse trees, and monadic second-order logic for matroids. J. Comb. Theory B 96(3), 325–351 (2006)CrossRefzbMATHGoogle Scholar
  31. 31.
    Kolaitis, P.G., Vardi, M.Y.: Conjunctive-query containment and constraint satisfaction. J. Comput. Sci. 61(2), 302–332 (2000)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Komusiewicz, C., Niedermeier, R.: New races in parameterized algorithmics. In: Proceedings of the 37th MFCS, LNCS, vol. 7464, pp. 19–30. Springer, Berlin (2012)Google Scholar
  33. 33.
    Lagergren, J., Arnborg, S.: Finding minimal forbidden minors using a finite congruence. In: Proceedings of the 18th ICALP, LCNS, vol. 510, pp. 532–543. Springer, Berlin (1991)Google Scholar
  34. 34.
    Lakshmipathy, N., Winklmann, K.: “Global” graph problems tend to be intractable. J. Comput. Syst. Sci. 32(3), 407–428 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Mahajan, S., Peters, J.G.: Regularity and locality in \(k\)-terminal graphs. Discrete Appl. Math. 54(2–3), 229–250 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Marx, D.: Approximating fractional hypertree width. ACM Trans Algorithms 6(2), 29 (2010)CrossRefMathSciNetGoogle Scholar
  37. 37.
    Miller, Z., Sudborough, I.H.: A polynomial algorithm for recognizing bounded cutwidth in hypergraphs. Math. Syst. Theory 24(1), 11–40 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Myhill, J.: Finite Automata and Representation of Events. Tech. Rep. WADD TR-57-624, Wright-Patterson Air Force Base, Ohio, USA (1957)Google Scholar
  39. 39.
    Nagamochi, H.: Linear layouts in submodular systems. In: Proceedings of the 23rd ISAAC, LNCS, vol. 7676, pp. 475–484. Springer, Berlin (2012)Google Scholar
  40. 40.
    Nerode, A.: Linear automaton transformations. Proc. Am. Math. Soc. 9(4), 541–544 (1958)CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  42. 42.
    Niedermeier, R.: Reflections on multivariate algorithmics and problem parameterization. In: Proceedings of the 27th STACS, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, LIPIcs, vol. 5, pp. 17–32 (2010)Google Scholar
  43. 43.
    Prasad, M.R., Chong, P., Keutzer, K.: Why is ATPG easy? In: Proceedings of the 36th DAC, pp. 22–28. ACM (1999)Google Scholar
  44. 44.
    Samer, M., Szeider, S.: Constraint satisfaction with bounded treewidth revisited. J. Comput. Sci. 76(2), 103–114 (2010)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Thilikos, D.M., Serna, M.J., Bodlaender, H.L.: Cutwidth I: a linear time fixed parameter algorithm. J. Algorithms 56(1), 1–24 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  46. 46.
    Wang, D., Clarke, E., Zhu, Y., Kukula, J.: Using cutwidth to improve symbolic simulation and Boolean satisfiability. In: Proceedings of the 6th HLDVT, pp. 165–170. IEEE (2001)Google Scholar
  47. 47.
    Wimer, T.V.: Linear Algorithms on \(k\)-Terminal Graphs. PhD thesis, Clemson University (1987)Google Scholar
  48. 48.
    Wimer, T.V., Hedetniemi, S.T., Laskar, R.: A methodology for constructing linear graph algorithms. Congr. Numer. 50, 43–60 (1985)MathSciNetGoogle Scholar
  49. 49.
    Yao, A.C.: Some complexity questions related to distributed computing. In: Proceedings of the 11th STOC, pp. 209–213. ACM (1979)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • René van Bevern
    • 1
    Email author
  • 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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