Constant girth approximation for directed graphs in subquadratic time

Shiri Chechik, Yang P. Liu, Omer Rotem, Aaron Sidford

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

In this paper we provide a Õ(mgn) time algorithm that computes a 3-multiplicative approximation of the girth of a n-node m-edge directed graph with non-negative edge lengths. This is the first algorithm which approximates the girth of a directed graph up to a constant multiplicative factor faster than All-Pairs Shortest Paths (APSP) time, i.e. O(mn). Additionally, for any integer k ≥ 1, we provide a deterministic algorithm for a O(kloglogn)-multiplicative approximation to the girth in directed graphs in Õ(m1+1/k) time. Combining the techniques from these two results gives us an algorithm for a O(klogk)-multiplicative approximation to the girth in directed graphs in Õ(m1+1/k) time. Our results naturally also provide algorithms for improved constructions of roundtrip spanners, the analog of spanners in directed graphs. The previous fastest algorithms for these problems either ran in All-Pairs Shortest Paths (APSP) time, i.e. O(mn), or were due Pachocki which provided a randomized algorithm that for any integer k ≥ 1 in time Õ(m1+1/k) computed with high probability a O(klogn) multiplicative approximation of the girth. Our first algorithm constitutes the first sub-APSP-time algorithm for approximating the girth to constant accuracy, our second removes the need for randomness and improves the approximation factor in Pachocki, and our third is the first time versus quality trade-off for obtaining constant approximations.

Original languageEnglish
Title of host publicationSTOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
EditorsKonstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy
PublisherAssociation for Computing Machinery
Pages1010-1023
Number of pages14
ISBN (Electronic)9781450369794
DOIs
StatePublished - 8 Jun 2020
Event52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 - Chicago, United States
Duration: 22 Jun 202026 Jun 2020

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020
Country/TerritoryUnited States
CityChicago
Period22/06/2026/06/20

Keywords

  • Girth
  • Graphs
  • Spanners

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