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Approximately Counting Embeddings into Random Graphs

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Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques (APPROX 2008, RANDOM 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5171))

Abstract

Let H be a graph, and let C H (G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C H (G). Previous results cover only a few specific instances of this general problem, for example, the case when H has degree at most one (monomer-dimer problem). In this paper, we present the first general subcase of the subgraph isomorphism counting problem which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the new decomposition is a labeling of the vertices generating a sequence of bipartite graphs. The decomposition permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this, we present a simple randomized algorithm for the counting problem. For all decomposable graphs H and all graphs G, the algorithm is an unbiased estimator. Furthermore, for all graphs H having a decomposition where each of the bipartite graphs generated is small and almost all graphs G, the algorithm is a fully polynomial randomized approximation scheme.

We show that the graph classes of H for which we obtain a fully polynomial randomized approximation scheme for almost all G includes graphs of degree at most two, bounded-degree forests, bounded-width grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs, whereas unbounded-width grid graphs are excluded.

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References

  1. Dong, H., Wu, Y., Ding, X.: An ARG representation for chinese characters and a radical extraction based on the representation. In: International Conference on Pattern Recognition, pp. 920–922 (1988)

    Google Scholar 

  2. Artymiuk, P.J., Bath, P.A., Grindley, H.M., Pepperrell, C.A., Poirrette, A.R., Rice, D.W., Thorner, D.A., Wild, D.J., Willett, P., Allen, F.H., Taylor, R.: Similarity searching in databases of three-dimensional molecules and macromolecules. Journal of Chemical Information and Computer Sciences 32, 617–630 (1992)

    Article  Google Scholar 

  3. Stahs, T., Wahl, F.M.: Recognition of polyhedral objects under perspective views. Computers and Artificial Intelligence 11, 155–172 (1992)

    MATH  MathSciNet  Google Scholar 

  4. Levinson, R.: Pattern associativity and the retrieval of semantic networks. Computers & Mathematics with Applications 23(6–9), 573–600 (1992)

    Article  MATH  Google Scholar 

  5. Alon, N., Yuster, R., Zwick, U.: Color-coding. Journal of the ACM 42(4), 844–856 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8, 189–201 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  7. Toda, S.: On the computational power of PP and ⊕ P. In: FOCS, pp. 514–519. IEEE, Los Alamitos (1989)

    Google Scholar 

  8. Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM Journal of Computing 33(4), 892–922 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Arvind, V., Raman, V.: Approximation algorithms for some parameterized counting problems. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 453–464. Springer, Heidelberg (2002)

    Google Scholar 

  10. Kasteleyn, P.W.: Graph theory and crystal physics. In: Harary, F. (ed.). Academic Press, London (1967)

    Google Scholar 

  11. Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of ACM 51(4), 671–697 (2004)

    Article  MathSciNet  Google Scholar 

  12. Jerrum, M., Sinclair, A.: Polynomial-time approximation algorithms for the Ising model. SIAM Journal on Computing 22(5), 1087–1116 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bezáková, I., Bhatnagar, N., Vigoda, E.: Sampling binary contingency tables with a greedy start. In: SODA, pp. 414–423. SIAM, Philadelphia (2006)

    Chapter  Google Scholar 

  14. Alon, N., Frieze, A.M., Welsh, D.: Polynomial time randomized approximation schemes for Tutte-Gröthendieck invariants: The dense case. Random Structures & Algorithms 6(4), 459–478 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dyer, M., Frieze, A.M., Jerrum, M.: Approximately counting Hamilton paths and cycles in dense graphs. SIAM Journal on Computing 27(5), 1262–1272 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jerrum, M.: Counting, sampling and integrating: algorithms and complexity. Birkhäuser, Basel (2003)

    MATH  Google Scholar 

  17. Frieze, A.M., McDiarmid, C.: Algorithmic theory of random graphs. Random Structures & Algorithms 10(1-2), 5–42 (1997)

    Article  MathSciNet  Google Scholar 

  18. Erdős, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–61 (1960)

    Google Scholar 

  19. Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM Journal on Computing 18(6), 1149–1178 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  20. Frieze, A.M., Jerrum, M.: An analysis of a Monte Carlo algorithm for estimating the permanent. Combinatorica 15(1), 67–83 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rasmussen, L.E.: Approximating the permanent: A simple approach. Random Structures & Algorithms 5(2), 349–362 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Chien, S.: A determinant-based algorithm for counting perfect matchings in a general graph. In: SODA, pp. 728–735. SIAM, Philadelphia (2004)

    Google Scholar 

  23. Fürer, M., Kasiviswanathan, S.P.: Approximately counting perfect matchings in general graphs. In: ALENEX/ANALCO, pp. 263–272. SIAM, Philadelphia (2005)

    Google Scholar 

  24. Frieze, A.M., Suen, S.: Counting the number of Hamilton cycles in random digraphs. Random Structures & Algorithms 3(3), 235–242 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. Frieze, A.M., Jerrum, M., Molloy, M.K., Robinson, R., Wormald, N.C.: Generating and counting Hamilton cycles in random regular graphs. Journal of Algorithms 21(1), 176–198 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rasmussen, L.E.: Approximately counting cliques. Random Structures & Algorithms 11(4), 395–411 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  27. Hammersley, J.M.: Existence theorems and Monte Carlo methods for the monomer-dimer problem. Research Papers in Statistics, 125–146 (1966)

    Google Scholar 

  28. Knuth, D.E.: Estimating the efficiency of backtrack programs. Mathematics of Computation 29(129), 121–136 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  29. Fürer, M., Kasiviswanathan, S.P.: An almost linear time approximation algorithm for the permanent of a random (0-1) matrix. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 263–274. Springer, Heidelberg (2004)

    Google Scholar 

  30. Sankowski, P.: Alternative algorithms for counting all matchings in graphs. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 427–438. Springer, Heidelberg (2003)

    Google Scholar 

  31. Jerrum, M., Valiant, L., Vazirani, V.: Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science 43, 169–188 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  32. Luks, E.M.: Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer and System Sciences 25, 42–65 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  33. Diestel, R.: Graph theory, 2nd edn. Springer, Heidelberg (2000)

    Google Scholar 

  34. Fürer, M., Kasiviswanathan, S.P.: Approximately counting embeddings into random graphs. CoRR, arXiv:0806.2287 [cs.DS] (2008)

    Google Scholar 

  35. Riordan, O.: Spanning subgraphs of random graphs. Combinatorics, Probability & Computing 9(2), 125–148 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  36. Janson, S., Łuczak, T., Ruciński, A.: Random graphs. Wiley-Interscience, Chichester (2000)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Computer Science and Engineering, Pennsylvania State University,  

    Martin Fürer & Shiva Prasad Kasiviswanathan

Authors
  1. Martin Fürer
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  2. Shiva Prasad Kasiviswanathan
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Ashish Goel Klaus Jansen José D. P. Rolim Ronitt Rubinfeld

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Fürer, M., Prasad Kasiviswanathan, S. (2008). Approximately Counting Embeddings into Random Graphs. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2008 2008. Lecture Notes in Computer Science, vol 5171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85363-3_33

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  • DOI: https://doi.org/10.1007/978-3-540-85363-3_33

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85362-6

  • Online ISBN: 978-3-540-85363-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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