## Abstract

Let G=(V,E) be an unweighted undirected graph with n vertices and m edges. Dor, Halperin, and Zwick [FOCS 1996, SICOMP 2000] presented an (min{n3/2m1/2,n7/3 })-time algorithm that computes estimated distances with an additive approximation of 2 without using Fast Matrix Multiplication (FMM). Recently, Deng, Kirkpatrick, Rong, V. Williams and Zhong [ICALP 2022] improved the running time for dense graphs to (n2.29)-time, using FMM, where an exact solution can be computed with FMM in (nω) time (ω < 2.37286) using Seidel's algorithm. Since an additive 2 approximation is also a multiplicative 2 approximation, computing an additive 2 approximation is at least as hard as computing a multiplicative 2 approximation. Thus, computing a multiplicative 2 approximation might be an easier problem. Nevertheless, more than two decades after the paper of Dor, Halperin, and Zwick was first published, no faster algorithm for computing multiplicative 2 approximation in dense graphs is known, rather then simply computing an additive 2 approximation. In this paper we present faster algorithms for computing a multiplicative 2 approximation without FMM. We show that in (min{ n1/2m ,n9/4 }) time it is possible to compute a multiplicative 2 approximation. For distances at least 4 we can get an even faster algorithm that in (min{ n7/4m1/4,n11/5}) expected time computes a multiplicative 2 approximation. Our algorithms are obtained by a combination of new ideas that take advantage of a careful new case analysis of the additive approximation algorithms of Dor, Halperin, and Zwick. More specifically, one of the main technical contributions we made is an analysis of the algorithm of Dor, Halperin, and Zwick that reveals certain cases in which their algorithm produces improved additive approximations without any modification. This analysis provides a full characterization of the instances for which it is harder to obtain an improved approximation. Using more ideas we can take care of some of these harder cases and to obtain an improved additive approximation also for them. Our new analysis, therefore, serves as a starting point for future research either on improved upper bounds or on conditional lower bounds.

Original language | English |
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Title of host publication | STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing |

Editors | Barna Saha, Rocco A. Servedio |

Publisher | Association for Computing Machinery |

Pages | 309-320 |

Number of pages | 12 |

ISBN (Electronic) | 9781450399135 |

DOIs | |

State | Published - 2 Jun 2023 |

Event | 55th Annual ACM Symposium on Theory of Computing, STOC 2023 - Orlando, United States Duration: 20 Jun 2023 → 23 Jun 2023 |

### Publication series

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

### Conference

Conference | 55th Annual ACM Symposium on Theory of Computing, STOC 2023 |
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Country/Territory | United States |

City | Orlando |

Period | 20/06/23 → 23/06/23 |

### Bibliographical note

Publisher Copyright:© 2023 Owner/Author.

## Keywords

- all pairs approximate shortest paths
- graph algorithms
- shortest paths