Approximately counting embeddings into random graphs

Martin Fürer, Shiva Prasad Kasiviswanathan

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    3 Scopus citations

    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.

    Original languageEnglish (US)
    Title of host publicationApproximation, Randomization and Combinatorial Optimization
    Subtitle of host publicationAlgorithms and Techniques - 11th International Workshop, APPROX 2008 and 12th International Workshop, RANDOM 2008, Proceedings
    Pages416-429
    Number of pages14
    DOIs
    StatePublished - Sep 22 2008
    Event11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008 - Boston, MA, United States
    Duration: Aug 25 2008Aug 27 2008

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume5171 LNCS
    ISSN (Print)0302-9743
    ISSN (Electronic)1611-3349

    Other

    Other11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008
    CountryUnited States
    CityBoston, MA
    Period8/25/088/27/08

    All Science Journal Classification (ASJC) codes

    • Theoretical Computer Science
    • Computer Science(all)

    Fingerprint Dive into the research topics of 'Approximately counting embeddings into random graphs'. Together they form a unique fingerprint.

    Cite this