## 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 +)-approximation (with high probability) for C_{k}, whose expected query complexity and running time are O _{Ck} ^{n} _{1}/_{k} + ^{m} _{C} ^{k} _{k} ^{/2} poly(log n, 1/, k). Hence, the complexity of the algorithm is sublinear in the size of the graph for C_{k} = (m^{k}/2−^{1}). Furthermore, the query complexity of our algorithm is essentially optimal (up to the dependence on log n, 1/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 for triangle counting, and does not generalize to all k. 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.

Original language | English |
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Title of host publication | STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Monika Henzinger, David Kempe, Ilias Diakonikolas |

Publisher | Association for Computing Machinery |

Pages | 100-113 |

Number of pages | 14 |

ISBN (Electronic) | 9781450355599 |

DOIs | |

State | Published - 20 Jun 2018 |

Externally published | Yes |

Event | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States Duration: 25 Jun 2018 → 29 Jun 2018 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 |
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Country/Territory | United States |

City | Los Angeles |

Period | 25/06/18 → 29/06/18 |

### Bibliographical note

Publisher Copyright:© 2018 Association for Computing Machinery.

## Keywords

- Approximation algorithms
- Counting cliques
- Sublinear algorithms