Approximately counting embeddings into random graphs

Martin Fürer, Shiva Prasad Kasiviswanathan

    Research output: Contribution to journalArticle

    1 Scopus citations

    Abstract

    Let H be a graph, and let CH(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating CH(G). Previous results cover only a few specific instances of this general problem, for example the case when H has degree at most one (the 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 decomposition is a labelling of the vertices such that every edge is between vertices with different labels, and for every vertex all neighbours with a higher label have identical labels. The labelling implicitly generates a sequence of bipartite graphs, which permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, 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, boundeddegree forests, bounded-width grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs of large girth, whereas unbounded-width grid graphs are excluded. Moreover, our general technique can easily be applied to proving many more similar results.

    Original languageEnglish (US)
    Pages (from-to)1028-1056
    Number of pages29
    JournalCombinatorics Probability and Computing
    Volume23
    Issue number6
    DOIs
    StatePublished - Nov 2 2014

    All Science Journal Classification (ASJC) codes

    • Theoretical Computer Science
    • Statistics and Probability
    • Computational Theory and Mathematics
    • Applied Mathematics

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