TY - GEN

T1 - Constant girth approximation for directed graphs in subquadratic time

AU - Chechik, Shiri

AU - Liu, Yang P.

AU - Rotem, Omer

AU - Sidford, Aaron

N1 - Publisher Copyright:
© 2020 ACM.

PY - 2020/6/8

Y1 - 2020/6/8

N2 - 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.

AB - 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.

KW - Girth

KW - Graphs

KW - Spanners

UR - http://www.scopus.com/inward/record.url?scp=85086767552&partnerID=8YFLogxK

U2 - 10.1145/3357713.3384330

DO - 10.1145/3357713.3384330

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AN - SCOPUS:85086767552

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 1010

EP - 1023

BT - STOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

A2 - Makarychev, Konstantin

A2 - Makarychev, Yury

A2 - Tulsiani, Madhur

A2 - Kamath, Gautam

A2 - Chuzhoy, Julia

PB - Association for Computing Machinery

T2 - 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020

Y2 - 22 June 2020 through 26 June 2020

ER -