## Abstract

The fastest known LP solver for general (dense) linear programs is due to [Cohen, Lee and Song'19] and runs in O?(n? +n2.5-?/2 + n2+1/6) time. A number of follow-up works [Lee, Song and Zhang'19, Brand'20, Song and Yu'20] obtain the same complexity through different techniques, but none of them can go below n2+1/6, even if ?=2. This leaves a polynomial gap between the cost of solving linear systems (n?) and the cost of solving linear programs, and as such, improving the n2+1/6 term is crucial toward establishing an equivalence between these two fundamental problems. In this paper, we reduce the running time to O?(n? +n2.5-?/2 + n2+1/18) where ? and ? are the fast matrix multiplication exponent and its dual. Hence, under the common belief that ? ? 2 and ? ? 1, our LP solver runs in O?(n2.055) time instead of O?(n2.16).

Original language | English |
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Title of host publication | STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Samir Khuller, Virginia Vassilevska Williams |

Publisher | Association for Computing Machinery |

Pages | 823-832 |

Number of pages | 10 |

ISBN (Electronic) | 9781450380539 |

DOIs | |

State | Published - 15 Jun 2021 |

Externally published | Yes |

Event | 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy Duration: 21 Jun 2021 → 25 Jun 2021 |

### Publication series

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

### Conference

Conference | 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 |
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Country/Territory | Italy |

City | Virtual, Online |

Period | 21/06/21 → 25/06/21 |

### Bibliographical note

Publisher Copyright:© 2021 ACM.

## Keywords

- Convex optimization
- Dynamic data-structure
- Linear programming