## Abstract

The Strong Exponential Time Hypothesis and the OV-conjecture are two popular hardness assumptions used to prove a plethora of lower bounds, especially in the realm of polynomial-time algorithms. The OV-conjecture in moderate dimension States there is no > 0 for which an O(N^{2−ε}) poly(D) time algorithm can decide whether there is a pair of orthogonal vectors in a given set of size N that contains D-dimensional binary vectors. We strengthen the evidence for these hardness assumptions. In particular, we show that if the OV-conjecture fails, then two problems for which we are far from obtaining even tiny improvements over exhaustive search would have surprisingly fast algorithms. If the OV conjecture is false, then there is a fixed > 0 such that: (1) For all d and all large enough k, there is a randomized algorithm that takes O(n(1^{−ε})^{k}) time to solve the Zero-Weight-k-Clique and Min-Weight-k-Clique problems on d-hypergraphs with n vertices. As a consequence, the OV-conjecture is implied by the Weighted Clique conjecture. (2) For all c, the satisfiability of sparse TC^{1} circuits on n inputs (that is, circuits with cn wires, depth c log n, and negation, AND, OR, and threshold gates) can be computed in time O((2 −)^{n}).

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 |

Place of Publication | New York |

Publisher | Association for Computing Machinery, Inc |

Pages | 253-266 |

Number of pages | 14 |

ISBN (Electronic) | 978-1-4503-5559-9 |

DOIs | |

Publication status | Published - 20 Jun 2018 |

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

### 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 |

## Keywords

- Clique
- Fine-grained complexity
- OV
- Satisfiability
- Threshold circuits
- fine-grained complexity
- clique
- satisfiability
- threshold circuits