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].
Original language | English |
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Title of host publication | 29th International Symposium on Algorithms and Computation (ISAAC 2018) |
Editors | Wen-Lian Hsu, Der-Tsai Lee, Chung-Shou Liao |
Place of Publication | Dagstuhl, Germany |
Publisher | Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik |
Pages | 1-12 |
ISBN (Print) | 978-3-95977-094-1 |
DOIs | |
Publication status | Published - 2018 |
MoE publication type | A4 Article in a conference publication |
Event | International Symposium on Algorithms and Computation - Jiaoxi, Taiwan, Republic of China Duration: 16 Dec 2018 → 19 Dec 2018 Conference number: 29 |
Publication series
Name | Leibniz International Proceedings in Informatics (LIPIcs) |
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Publisher | Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik |
Volume | 123 |
ISSN (Electronic) | 1868-8969 |
Conference
Conference | International Symposium on Algorithms and Computation |
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Abbreviated title | ISAAC |
Country/Territory | Taiwan, Republic of China |
City | Jiaoxi |
Period | 16/12/2018 → 19/12/2018 |
Keywords
- graph embedding
- k-path
- subgraph counting
- maximum degree