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.
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Fürer, M., Prasad Kasiviswanathan, S. (2008). Approximately Counting Embeddings into Random Graphs. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2008 2008. Lecture Notes in Computer Science, vol 5171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85363-3_33
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DOI: https://doi.org/10.1007/978-3-540-85363-3_33
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