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International Conference on Theory and Applications of Models of Computation

TAMC 2009: Theory and Applications of Models of Computation pp 410–419Cite as

Greedy Local Search and Vertex Cover in Sparse Random Graphs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5532))

Abstract

Recently, various randomized search heuristics have been studied for the solution of the minimum vertex cover problem, in particular for sparse random instances according to the G(n,c/n) model, where c > 0 is a constant. Methods from statistical physics suggest that the problem is easy if c < e. This work starts with a rigorous explanation for this claim based on the refined analysis of the Karp-Sipser algorithm by Aronson et al. Subsequently, theoretical supplements are given to experimental studies of search heuristics on random graphs. For c < 1, a greedy and randomized local-search heuristic finds an optimal cover in polynomial time with a probability arbitrarily close to 1. This behavior relies on the absence of a giant component. As an additional insight into the randomized search, it is shown that the heuristic fails badly also on graphs consisting of a single tree component of maximum degree 3.

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References

  1. Aronson, J., Frieze, A., Pittel, B.G.: Maximum matchings in sparse random graphs: Karp-Sipser revisited. Random Structures and Algorithms 12(2), 111–177 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bauer, M., Golinelli, O.: Core percolation in random graphs: a critical phenomena analysis. The European Physical Journal B 24(3), 339–352 (2001)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  4. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)

    MATH  Google Scholar 

  5. Evans, I.K.: Evolutionary algorithms for vertex cover. In: Porto, V.W., Waagen, D. (eds.) EP 1998. LNCS, vol. 1447, pp. 377–386. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  6. Feller, W.: An Introduction to Probability Theory and Its Applications, 3rd edn., vol. 1. Wiley, Chichester (1968)

    MATH  Google Scholar 

  7. Friedrich, T., He, J., Hebbinghaus, N., Neumann, F., Witt, C.: Approximating covering problems by randomized search heuristics using multi-objective models. In: Proc. of GECCO 2007, pp. 797–804. AMC Press (2007)

    Google Scholar 

  8. Friedrich, T., He, J., Hebbinghaus, N., Neumann, F., Witt, C.: Analyses of simple hybrid evolutionary algorithms for the vertex cover problem. Evolutionary Computation 17(1), 3–20 (2009)

    Article  Google Scholar 

  9. Gamarnik, D., Nowicki, T., Swirscsz, G.: Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method. Random Structures and Algorithms 28(1), 76–106 (2005)

    Article  Google Scholar 

  10. Giel, O., Wegener, I.: Evolutionary algorithms and the maximum matching problem. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 415–426. Springer, Heidelberg (2003)

    Google Scholar 

  11. Hartmann, A., Weigt, M.: Statistical mechanics perspective on the phase transition in vertex covering of finite-connectivity random graphs. Theoretical Computer Science  (265), 199–225 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Karp, R.M., Sipser, M.: Maximum matchings in sparse random graphs. In: Proc. of FOCS 1981, pp. 364–375. IEEE Press, Los Alamitos (1981)

    Google Scholar 

  13. Neumann, F., Wegener, I.: Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. Theoretical Computer Science 378(1), 32–40 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Neumann, F., Witt, C.: Runtime analysis of a simple ant colony optimization algorithm. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 618–627. Springer, Heidelberg (2006); Extended version to appear in Algorithmica

    Chapter  Google Scholar 

  15. Oliveto, P.S., He, J., Yao, X.: Evolutionary algorithms and the vertex cover problem. In: Proc. of CEC 2007, pp. 1870–1877. IEEE Press, Los Alamitos (2007)

    Google Scholar 

  16. Pelikan, M., Kalapala, R., Hartmann, A.K.: Hybrid evolutionary algorithms on minimum vertex cover for random graphs. In: Proc. of GECCO 2007, pp. 547–554. ACM Press, New York (2007)

    Google Scholar 

  17. Tarjan, R.E., Trojanowski, A.E.: Finding a maximum independent set. SIAM Journal on Computing 6(3), 537–546 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  18. Wegener, I.: Simulated annealing beats metropolis in combinatorial optimization. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 589–601. Springer, Heidelberg (2005)

    Google Scholar 

  19. Witt, C.: Worst-case and average-case approximations by simple randomized search heuristics. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 44–56. Springer, Heidelberg (2005)

    Google Scholar 

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Author information

Authors and Affiliations

  1. DTU Informatics, Technical University of Denmark, 2800 Kgs., Lyngby, Denmark

    Carsten Witt

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  1. Carsten Witt
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Texas A&M University, College Station, 77843, Texas, USA

    Jianer Chen

  2. School of Mathematics, University of Leeds, LS2 9JT, U.K.

    S. Barry Cooper

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Witt, C. (2009). Greedy Local Search and Vertex Cover in Sparse Random Graphs. In: Chen, J., Cooper, S.B. (eds) Theory and Applications of Models of Computation. TAMC 2009. Lecture Notes in Computer Science, vol 5532. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02017-9_43

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  • DOI: https://doi.org/10.1007/978-3-642-02017-9_43

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02016-2

  • Online ISBN: 978-3-642-02017-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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