Approximately Counting Embeddings into Random Graphs

• Authors: Martin Fürer;Shiva Prasad Kasiviswanathan
• Venue: APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
• Year: 2008

Let Hbe a graph, and let CH(G) be the number of (subgraph isomorphic) copies of Hcontained 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 Hhas 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 Hand all graphs G, the algorithm is an unbiased estimator. Furthermore, for all graphs Hhaving 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 Hfor which we obtain a fully polynomial randomized approximation scheme for almost all Gincludes 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.