Approximately Counting Embeddings into Random Graphs

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 (monomerdimer 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 the same label. 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 p...

• Type: Conference
• Year: 2008
• Where: APPROX
• Authors: Martin Fürer, Shiva Prasad Kasiviswanathan
• Keywords: Algorithms, APPROX 2008, Bipartite Graphs, Counting Problem, Polynomial Randomized Approximation