Approximately Counting Embeddings into Random Graphs

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 decomposition is a labeling of the vertices such that every edge is between vertices with different labels and for every vertex all neighbors with a higher label have identical labels. The labeling 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, bounded-degree forests, bounded-length grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs, whereas unbounded-length grid graphs are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition 3.

Counting cliques and clique covers in random graphs

“…If the procedure succeeds, we compute the probability with which the clique is obtained in G and output its inverse. As shown in [12], this is an unbiased estimate of the number of cliques in G. We state the results below in Theorem 1. In this work, we generalize Rasmussen's approach [28] to efficiently count k-cliques and k-clique covers in random graphs.…”

“…Our algorithm is based on the idea of Rasmussen's unbiased estimator for permanents [28]. It has been widely used in the context of subgraph isomorphism counting problems [29,11,12]. For counting k-cliques in the input random graph G, we embed a k-clique into G, doing so one vertex at a time chosen randomly.…”

“…Various approaches for getting an unbiased estimator with small variance have been explored for counting perfect matchings in other graphs. Some of these are determinant based approaches [13,20,5,25], Markov chain Monte Carlo (MCMC) algorithms [3,17,18,2] and search based on Rasmussen's techniques [28,29,11,12]. Chien [5] gives an efficient fpras for counting perfect matchings in random graphs.…”

Fast Relational Probabilistic Inference and Learning: Approximate Counting via Hypergraphs

“…While these approaches are interesting with theoretical guarantees, they apply to specific classes of simple hypergraphs with bounded degree and regularity which cannot model real-world multi-relational data. Fürer and Kasiviswanathan (2014) propose a polynomial time sampling-based approximation strategy for counting isomorphic subgraphs matching a given template leveraging bounded-width ordered bipartite decompositions of the templates. A key feature of this approach is its provable generalization across varied classes of graphs.…”

Graph and Network Pattern Mining

“…E.g. for certain pattern classes, there exist efficient algorithms which provide good approximations on almost all random graphs [31] On the other hand, recent work on fixed parameter tractability has shown that there are algorithms, often randomized ones, whose assymptotic complexity is exponential in the pattern size but only polynomial (e.g. linear) in the network size.…”