research-article

On approximating the number of k-cliques in sublinear time

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C. SeshadhriAuthors Info & Claims

STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

Pages 722 - 734

https://doi.org/10.1145/3188745.3188810

Published: 20 June 2018 Publication History

Abstract

We study the problem of approximating the number of k-cliques in a graph when given query access to the graph. We consider the standard query model for general graphs via (1) degree queries, (2) neighbor queries and (3) pair queries. Let n denote the number of vertices in the graph, m the number of edges, and C_k the number of k-cliques. We design an algorithm that outputs a (1+ε)-approximation (with high probability) for C_k, whose expected query complexity and running time are O(n/C_k^1/k+m^k/2/C_k )(logn, 1/ε,k).

Hence, the complexity of the algorithm is sublinear in the size of the graph for C_k = ω(m^k/2−1). Furthermore, we prove a lower bound showing that the query complexity of our algorithm is essentially optimal (up to the dependence on logn, 1/ε and k).

The previous results in this vein are by Feige (SICOMP 06) and by Goldreich and Ron (RSA 08) for edge counting (k=2) and by Eden et al. (FOCS 2015) for triangle counting (k=3). Our result matches the complexities of these results.

The previous result by Eden et al. hinges on a certain amortization technique that works only for triangle counting, and does not generalize for larger cliques. We obtain a general algorithm that works for any k≥ 3 by designing a procedure that samples each k-clique incident to a given set S of vertices with approximately equal probability. The primary difficulty is in finding cliques incident to purely high-degree vertices, since random sampling within neighbors has a low success probability. This is achieved by an algorithm that samples uniform random high degree vertices and a careful tradeoff between estimating cliques incident purely to high-degree vertices and those that include a low-degree vertex.

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Index Terms

On approximating the number of k-cliques in sublinear time
1. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

June 2018

1332 pages

ISBN:9781450355599

DOI:10.1145/3188745

General Chairs:
Ilias Diakonikolas
University of Southern California, USA
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David Kempe
University of Southern California, USA
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Monika Henzinger
University of Vienna, Austria

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https://doi.org/10.1109/TKDE.2023.3314643
Bera SChoudhari JHaddadan SAhmadian SChua TLauw HSi LTerzi ETsaparas P(2023)DeMEtRIS: Counting (near)-Cliques by CrawlingProceedings of the Sixteenth ACM International Conference on Web Search and Data Mining10.1145/3539597.3570438(312-320)Online publication date: 27-Feb-2023
https://dl.acm.org/doi/10.1145/3539597.3570438
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