Counting Connected Subgraphs with Maximum-Degree-Aware Sieving

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Date
2018
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Language
en
Pages
1-12
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29th International Symposium on Algorithms and Computation (ISAAC 2018), Leibniz International Proceedings in Informatics (LIPIcs), Volume 123
Abstract
We study the problem of counting the isomorphic occurrences of a k-vertex pattern graph P as a subgraph in an n-vertex host graph G. Our specific interest is on algorithms for subgraph counting that are sensitive to the maximum degree Delta of the host graph. Assuming that the pattern graph P is connected and admits a vertex balancer of size b, we present an algorithm that counts the occurrences of P in G in O ((2 Delta-2)^{(k+b)/2} 2^{-b} n/(Delta) k^2 log n) time. We define a balancer as a vertex separator of P that can be represented as an intersection of two equal-size vertex subsets, the union of which is the vertex set of P, and both of which induce connected subgraphs of P. A corollary of our main result is that we can count the number of k-vertex paths in an n-vertex graph in O((2 Delta-2)^{floor[k/2]} n k^2 log n) time, which for all moderately dense graphs with Delta <= n^{1/3} improves on the recent breakthrough work of Curticapean, Dell, and Marx [STOC 2017], who show how to count the isomorphic occurrences of a q-edge pattern graph as a subgraph in an n-vertex host graph in time O(q^q n^{0.17q}) for all large enough q. Another recent result of Brand, Dell, and Husfeldt [STOC 2018] shows that k-vertex paths in a bounded-degree graph can be approximately counted in O(4^kn) time. Our result shows that the exact count can be recovered at least as fast for Delta<10. Our algorithm is based on the principle of inclusion and exclusion, and can be viewed as a sparsity-sensitive version of the "counting in halves"-approach explored by Björklund, Husfeldt, Kaski, and Koivisto [ESA 2009].
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Keywords
graph embedding, k-path, subgraph counting, maximum degree
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Citation
Björklund , A , Husfeldt , T , Kaski , P & Koivisto , M 2018 , Counting Connected Subgraphs with Maximum-Degree-Aware Sieving . in W-L Hsu , D-T Lee & C-S Liao (eds) , 29th International Symposium on Algorithms and Computation (ISAAC 2018) . , 17 , Leibniz International Proceedings in Informatics (LIPIcs) , vol. 123 , Schloss Dagstuhl - Leibniz-Zentrum für Informatik , Dagstuhl, Germany , pp. 1-12 , International Symposium on Algorithms and Computation , Jiaoxi , Taiwan, Republic of China , 16/12/2018 . https://doi.org/10.4230/LIPIcs.ISAAC.2018.17