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International Symposium on Algorithms and Computation

ISAAC 2005: Algorithms and Computation pp 493–504Cite as

Simple and Efficient Greedy Algorithms for Hamilton Cycles in Random Intersection Graphs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3827))

Abstract

In this work we consider the problem of finding Hamilton Cycles in graphs derived from the uniform random intersection graphs model G n, m, p . In particular, (a) for the case m = n α, α> 1 we give a result that allows us to apply (with the same probability of success) any algorithm that finds a Hamilton cycle with high probability in a G n, k graph (i.e. a graph chosen equiprobably form the space of all graphs with k edges), (b) we give an expected polynomial time algorithm for the case p = constant and \(m \leq \alpha {\sqrt{{n}\over {{\rm log}n}}}\) for some constant α, and (c) we show that the greedy approach still works well even in the case \(m = o({{n}\over{{\rm log}n}})\) and p just above the connectivity threshold of G n, m, p (found in [21]) by giving a greedy algorithm that finds a Hamilton cycle in those ranges of m, p with high probability.

This work has been partially supported by the IST Programme of the European Union under contract number 001907 (DELIS) and by the GSRT PENED 2003 ALGO.D.E.S. Project.

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References

  1. Alon, N., Spencer, J.H.: The Probabilistic Method, 2nd edn. John Wiley & Sons, Inc., Chichester (2000)

    Book  MATH  Google Scholar 

  2. Beier, R., Vöcking, B.: Random Knapsack in Expected Polynomial Time. In: The Proc. of the 35th Annual ACM Symposium on Theory of Computing, pp. 232–241. ACM Press, New York (2003)

    Google Scholar 

  3. Bollobás, B.: Random Graphs, 2nd edn. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  4. Bollobás, B., Fenner, T.I., Frieze, A.M.: An algorithm for Finding Hamilton Paths and Cycles in Random Graphs. Combinatorica 7, 327–341

    Google Scholar 

  5. Coja-Oghlan, A., Taraz, A.: Colouring Random Graphs in Expected Polynomial Time. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 487–498. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  6. Díaz, J., Penrose, M.D., Petit, J., Serna, M.: Approximating Layout Problems on Random Geometric Graphs. Journal of Algorithms 39, 78–116 (2001)

    Article  MATH  Google Scholar 

  7. Díaz, J., Petit, J., Serna, M.: A Random Graph Model for Optical Networks of Sensors. In: The 1st International Workshop on Efficient and Experimental Algorithms, WEA (2003); Also in the IEEE Transactions on Mobile Computing Journal 2(3), 186–196 (2003)

    Google Scholar 

  8. Díaz, J., Petit, J., Serna, M.: Random Geometric Problems on [0, 1]2. In: Rolim, J.D.P., Serna, M., Luby, M. (eds.) RANDOM 1998. LNCS, vol. 1518, pp. 294–306. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  9. Efthymiou, H., Spirakis, P.: On the Existence of Hamiltonian Cycles in Random Intersection Graphs. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 690–701. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  10. Fill, J.A., Sheinerman, E.R., Singer-Cohen, K.B.: Random Intersection Graphs when m = ω(n): An Equivalence Theorem Relating the Evolution of the G(n, m, p) and G(n, p) models, http://citeseer.nj.nec.com/fill98random.html

  11. Godehardt, E., Jaworski, J.: Two models of Random Intersection Graphs for Classification. In: Opitz, O., Schwaiger, M. (eds.) Studies in Classification, Data Analysis and Knowledge Organisation, pp. 67–82. Springer, Heidelberg (2002)

    Google Scholar 

  12. Gurevich, Y., Shelah, S.: Expected Computation Time for Hamiltonian Path Problem. SIAM Journal on Computing 16, 486–502 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. Karoński, M., Scheinerman, E.R., Singer- Cohen, K.B.: On Random Intersection Graphs: The Subgraph Problem. Combinatorics, Probability and Computing journal 8, 131–159 (1999)

    Article  MATH  Google Scholar 

  14. Marczewski, E.: Sur deux propriétés des classes d’ ensembles. Fund. Math. 33, 303–307 (1945)

    MATH  MathSciNet  Google Scholar 

  15. Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  16. Nikoletseas, S., Palem, K., Spirakis, P., Yung, M.: Short Vertex Disjoint Paths and Multiconnectivity in Random Graphs: Reliable Network Computing. In: Shamir, E., Abiteboul, S. (eds.) ICALP 1994. LNCS, vol. 820, pp. 247–262. Springer, Heidelberg (1994)

    Google Scholar 

  17. Nikoletseas, S., Raptopoulos, C., Spirakis, P.: The Existence and Efficient Construction of Large Independent Sets in General Random Intersection Graphs. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1029–1040. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  18. Nikoletseas, S., Spirakis, P.: Expander Properties in Random Regular Graphs with Edge Faults. In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, pp. 421–432. Springer, Heidelberg (1995)

    Google Scholar 

  19. Penrose, M.: Random Geometric Graphs. Oxford Studies in Probability (2003)

    Google Scholar 

  20. Raptopoulos, C., Spirakis, P.: Simple and Efficient Greedy Algorithms for Hamilton Cycles in Random Intersection Graphs. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 493–504. Springer, Heidelberg (2005), http://students.ceid.upatras.gr/~raptopox/fullpaper.ps

    Chapter  Google Scholar 

  21. Singer-Cohen, K.B.: Random Intersection Graphs. PhD thesis, John Hopkins University (1995)

    Google Scholar 

  22. Thomason, A.: A simple linear expected time algorithm for the Hamilton cycle problem. Discrete Math. 75, 373–379 (1989)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Computer Technology Institute, P.O. Box 1122, 26110, Patras, Greece

    C. Raptopoulos & P. Spirakis

  2. University of Patras, 26500, Patras, Greece

    C. Raptopoulos & P. Spirakis

Authors
  1. C. Raptopoulos
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  2. P. Spirakis
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Editor information

Editors and Affiliations

  1. Department of Computer Science, City University of Hong Kong, Tat Chee Avenue, Kowloon , Hong Kong

    Xiaotie Deng

  2. Department of Computer Science, University of Texas at Dallas, EE/CS Building, 75083, Richardson, TX, USA

    Ding-Zhu Du

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Raptopoulos, C., Spirakis, P. (2005). Simple and Efficient Greedy Algorithms for Hamilton Cycles in Random Intersection Graphs. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_50

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  • DOI: https://doi.org/10.1007/11602613_50

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30935-2

  • Online ISBN: 978-3-540-32426-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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