Testing hereditary properties efficiently on average

  • Jens Gustedt
  • Angelika Steger
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 831)


We use the quasi-ordering of substructure relations such as induced and weak subgraph, induced suborder, graph minor or subformula of a CNF formula to obtain recognition algorithms for hereditary properties that are fast on average. The ingredients needed besides inheritance are independence of the occurrence of small substructures in a random input and the existence of algorithms for recognition that are at most exponential.


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  1. [ALS91]
    Stefan Arnborg, Jens Lagergren, and Detlef Seese, Easy problems for tree-decomposable graphs, J. Algorithms 12 (1991), 308–340.Google Scholar
  2. [BFF85]
    B. Bollobás, T.I. Fenner, and A.M. Frieze, An algorithm for finding hamilton paths and cycles in random graphs, in [STO85] (1985), 430–439.Google Scholar
  3. [BK79]
    L. Babai and L. Kucera, Canonical labelling of graphs in linear time, in [FOC79] (1979), 39–46.Google Scholar
  4. [Bod93]
    Hans L. Bodlaender, A linear time algorithm for finding tree-decompositions of small treewidth, in [STO93] (1993).Google Scholar
  5. [Bol85]
    Belá Bollobás, Random graphs, Academic Press, London, 1985.Google Scholar
  6. [Com88]
    K. J. Compton, The computational complexity of asymptotic problems. I: Partial orders, Inform. and Complexity 78 (1988), 103–123.Google Scholar
  7. [DF89]
    M.E. Dyer and A.M. Frieze, The solution of some random NP-hard problems in polynomial expected time, J. Algorithms 10 (1989), 451–489.Google Scholar
  8. [ER60]
    P. Erdős and A. Rényi, On the evolution of random graphs, Madyar Tnd. Akad. Mat. Kut. Int. Kőzl. 6 (1960), 17–61.Google Scholar
  9. [Fel89]
    Michael R. Fellows, The Robertson-Seymour theorems: a survey of applications, see [GA89], 1989, pp. 1–18.Google Scholar
  10. [FL85]
    Michael R. Fellows and Michael A. Langston, Nonconstructive advances in polynomial-time complexity, Inform. Process. Lett. (1985).Google Scholar
  11. [FL88]
    Michael R. Fellows and Michael A. Langston, Nonconstructive tools for proving polynomial-time decidability, J. Assoc. Comput. Mach. 35 (1988), no. 3, 727–739.Google Scholar
  12. [FL92]
    Michael R. Fellows and Michael A. Langston, On well-partial-order theory and its application to combinatorial problems of VLSI design, SIAM J. Disc. Math. 5 (1992), no. 1, 117–126.Google Scholar
  13. [FOC79]
    20th Annual Symposion On Foundations of Computer Science, IEEE, The Institute of Electrical and Electronics Engineers, IEEE Computer Society Press, 1979.Google Scholar
  14. [FP83]
    John Franco and Marvin Paull, Probabilistic analysis of the Dams Putnam procedure for solving the satisfiability problem, Discrete Appl. Math. 5 (1983), 77–87.Google Scholar
  15. [Fri90]
    A.M. Frieze, Probabilistic analysis of graph algorithms, Computing 7 (1990), 209–233.Google Scholar
  16. [GA89]
    Graphs and Algorithms (R. B. Richter, ed.), American Math. Soc., Contemp. Math., 89, 1989, Proceedings of the AMS-IMS-SIAM joint Summer Research Conference 1987.Google Scholar
  17. [GJ79]
    Michael R. Garey and David S. Johnson, Computers and Intractability, W. H. Freeman and Company, New York, 1979.Google Scholar
  18. [Gol80]
    Martin C. Golumbic, Algorithmic graph theory and perfect graphs, Academic Press, London, New York, 1980.Google Scholar
  19. [GPB82]
    A. Goldberg, P. Purdom, and C. Brown, Average time analysis of simplified Davis-Putnam procedures, Inform. Process. Lett. 15 (1982), 72–75, Corrigendum, Inform. Process. Lett. 16 (1982), 213.Google Scholar
  20. [Gus92]
    Jens Gustedt, Algorithmic aspects of ordered structures, Ph.D. thesis, Technische Universität Berlin, 1992.Google Scholar
  21. [HTL91]
    T. H. Hu, C. Y. Tang, and R. C. T. Lee, An average case analysts of Monien and Speckmeyer's mechanical theorem proving algorithm, in [ISA91] (1991), 116–126.Google Scholar
  22. [ISA91]
    ISA '91 Algorithms (Wen-Lian Hsu and R.C.T. Lee, eds.), Springer-Verlag, 1991, LNCS 557, Proceedings of the 2nd International Symposion on Algorithms, Taipei, Republic of China.Google Scholar
  23. [Iwa89]
    Kazuo Iwama, CNF satisfiability test by counting and polynomial average time, SIAM J. Comput. 18 (1989), no. 2, 385–391.Google Scholar
  24. [JLR90]
    Svante Janson, Thomasz Luczak, and Andrezj Ruciński, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, Random Graphs '87 (M. Karoński, J. Jaworski, and A. Ruciński, eds.), John Wiley & Sons, New York, 1990, pp. 73–87.Google Scholar
  25. [Joh84]
    David S. Johnson, The NP-completeness column: An ongoing guide, “Solving NP-hard Problems Quickly (On Average)”, J. Algorithms 5 (1984), 284–299.Google Scholar
  26. [Möh90]
    Rolf H. Möhring, Graph problems related to gate matrix layout and PLA folding, Computational Graph Theory (Wien) (G. Tinhofer et al., eds.), Springer-Verlag, Wien, 1990, pp. 17–52.Google Scholar
  27. [PS92a]
    Hans Jürgen Prömel and Angelika Steger, Coloring clique-free graphs in linear expected time, Random Structures and Algorithms 3 (1992), 374–402.Google Scholar
  28. [PS92b]
    Hans Jürgen Prömel and Angelika Steger, Excluding induced subgraphs III: A general asymtotic, Random Structures and Algorithms 3 (1992), no. 1, 19–31.Google Scholar
  29. [RS85a]
    Neil Robertson and Paul Seymour, Disjoint paths — a survey, SIAM J. Algebraic Discrete Methods 6 (1985), no. 2, 300–305.Google Scholar
  30. [RS85b]
    Neil Robertson and Paul Seymour, Graph minors — a survey, Surveys in Combinatorics (Glasgow 1985) (I. Anderson, ed.), Cambridge Univ. Press, Cambridge-New York, 1985, pp. 153–171.Google Scholar
  31. [STO85]
    Proceedings of the Seventeenth Anual ACM Symposion on Theory of Computing, ACM, Assoc. for Comp. Machinery, 1985.Google Scholar
  32. [STO93]
    Proceedings of the Twenty Fifth Anual ACM Symposion on Theory of Computing, ACM, Assoc. for Comp. Machinery, 1993.Google Scholar
  33. [Tho84]
    A.G. Thomason, An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95 (1984), 261–265.Google Scholar
  34. [Wil84]
    H. S. Wilf, Backtrack: an O(1) texpected time algorithm for the graph coloring problem, Inform. Process. Lett. 18 (1984), 119–121.Google Scholar
  35. [Win85]
    Peter Winkler, Random orders, Order 1 (1985), 317–331.Google Scholar
  36. [Win89]
    Peter Winkler, A counterexample in the theory of random orders, Order 5 (1989), 363–368.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Jens Gustedt
    • 1
  • Angelika Steger
    • 2
  1. 1.Fachbereich MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Forschungsinstitut für Diskrete MathematikUniversität BonnBonnGermany

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