## Abstract

The 3SUM conjecture has proven to be a valuable tool for proving conditional lower bounds on dynamic data structures and graph problems. This line of work was initiated by Patrascu (STOC 2010) who reduced 3SUM to an offline SetDisjointness problem. However, the reduction introduced by Patrascu suffers from several inefficiencies, making it difficult to obtain tight conditional lower bounds from the 3SUM conjecture. In this paper we address many of the deficiencies of Patrascu's framework. We give new and efficient reductions from 3SUM to offline SetDisjointness and offline Setlntersection (the reporting version of SetDisjointness) which leads to polyno-mially higher lower bounds on several problems. Using our reductions, we are able to show the essential optimality of several algorithms, assuming the 3SUM conjecture. Chiba and Nishizeki's O(ma)-time algorithm (SICOMP 1985) for enumerating all triangles in a graph with arboricity/degeneracy α is essentially optimal, for any α. Bjorklund, Pagh, Williams, and Zwick's algorithm (ICALP 2014) for listing t triangles is essentially optimal (assuming the matrix multiplication exponent is ω = 2). Any static data structure for SetDisjointness that answers queries in constant time must spend n{N2-o^) time in preprocessing, where N is the size of the set system. These statements were unattainable via Patrascu's reductions. We also introduce several new reductions from 3SUM to pattern matching problems and dynamic graph problems. Of particular interest are new conditional lower bounds for dynamic versions of Maximum Cardinality Matching, which introduce a new technique for obtaining amortized lower bounds.

Original language | English |
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Title of host publication | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |

Editors | Robert Krauthgamer |

Publisher | Association for Computing Machinery |

Pages | 1272-1287 |

Number of pages | 16 |

ISBN (Electronic) | 9781510819672 |

State | Published - 2016 |

Event | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 - Arlington, United States Duration: 10 Jan 2016 → 12 Jan 2016 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 2 |

### Conference

Conference | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |
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Country/Territory | United States |

City | Arlington |

Period | 10/01/16 → 12/01/16 |

### Bibliographical note

Publisher Copyright:© Copyright (2016) by SIAM: Society for Industrial and Applied Mathematics.