## Abstract

In this paper we give sublinear-time distributed algorithms in the CONGEST model for subgraph detection for two classes of graphs: cliques and even-length cycles. We show for the first time that all copies of 4-cliques and 5-cliques in the network graph can be listed in sublinear time, O(n^{5/6+o(1)}) rounds and O(n^{21/22+o(1)}) rounds, respectively. Prior to our work, it was not known whether it was possible to even check if the network contains a 4-clique or a 5-clique in sublinear time. For even-length cycles, C_{2k}, we give an improved sublinear-time algorithm, which exploits a new connection to extremal combinatorics. For example, for 6-cycles we improve the running time from Õ(n^{5/6}) to Õ(n^{3/4}) rounds. We also show two obstacles on proving lower bounds for C_{2k}-freeness: First, we use the new connection to extremal combinatorics to show that the current lower bound of Ω(√n) rounds for 6-cycle freeness cannot be improved using partition-based reductions from 2-party communication complexity, the technique by which all known lower bounds on subgraph detection have been proven to date. Second, we show that there is some fixed constant δ ∈ (0,1/2) such that for any k, a Ω(n^{1/2+δ}) lower bound on C_{2k}-freeness implies new lower bounds in circuit complexity. For general subgraphs, it was shown in [14] that for any fixed k, there exists a subgraph H of size k such that H-freeness requires Ω(n^{2−Θ(1/k})) rounds. It was left as an open problem whether this is tight, or whether some constant-sized subgraph requires truly quadratic time to detect. We show that in fact, for any subgraph H of constant size k, the H-freeness problem can be solved in O(n^{2−Θ(1/k)}) rounds, nearly matching the lower bound of [14].

Original language | English |
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Title of host publication | 33rd International Symposium on Distributed Computing, DISC 2019 |

Editors | Jukka Suomela |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771269 |

DOIs | |

State | Published - Oct 2019 |

Externally published | Yes |

Event | 33rd International Symposium on Distributed Computing, DISC 2019 - Budapest, Hungary Duration: 14 Oct 2019 → 18 Oct 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 146 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 33rd International Symposium on Distributed Computing, DISC 2019 |
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Country/Territory | Hungary |

City | Budapest |

Period | 14/10/19 → 18/10/19 |

### Bibliographical note

Publisher Copyright:© Talya Eden, Nimrod Fiat, Orr Fischer, Fabian Kuhn, and Rotem Oshman.

## Keywords

- CONGEST
- Distributed computing
- Subgraph freeness