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Title: Approximating Min-Diameter: Standard and Bichromatic
The min-diameter of a directed graph G is a measure of the largest distance between nodes. It is equal to the maximum min-distance d_{min}(u,v) across all pairs u,v ∈ V(G), where d_{min}(u,v) = min(d(u,v), d(v,u)). Min-diameter approximation in directed graphs has attracted attention recently as an offshoot of the classical and well-studied diameter approximation problem. Our work provides a 3/2-approximation algorithm for min-diameter in DAGs running in time O(m^{1.426} n^{0.288}), and a faster almost-3/2-approximation variant which runs in time O(m^{0.713} n). (An almost-α-approximation algorithm determines the min-diameter to within a multiplicative factor of α plus constant additive error.) This is the first known algorithm to solve 3/2-approximation for min-diameter in sparse DAGs in truly subquadratic time O(m^{2-ε}) for ε > 0; previously only a 2-approximation was known. By a conditional lower bound result of [Abboud et al, SODA 2016], a better than 3/2-approximation can't be achieved in truly subquadratic time under the Strong Exponential Time Hypothesis (SETH), so our result is conditionally tight. We additionally obtain a new conditional lower bound for min-diameter approximation in general directed graphs, showing that under SETH, one cannot achieve an approximation factor below 2 in truly subquadratic time. Our work also presents the first study of approximating bichromatic min-diameter, which is the maximum min-distance between oppositely colored vertices in a 2-colored graph. We show that SETH implies that in DAGs, a better than 2 approximation cannot be achieved in truly subquadratic time, and that in general graphs, an approximation within a factor below 5/2 is similarly out of reach. We then obtain an O(m)-time algorithm which determines if bichromatic min-diameter is finite, and an almost-2-approximation algorithm for bichromatic min-diameter with runtime Õ(min(m^{4/3} n^{1/3}, m^{1/2} n^{3/2})).  more » « less
Award ID(s):
2129139
PAR ID:
10565793
Author(s) / Creator(s):
; ;
Editor(s):
Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J; Herman, Grzegorz
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
274
ISSN:
1868-8969
ISBN:
978-3-95977-295-2
Page Range / eLocation ID:
274-274
Subject(s) / Keyword(s):
diameter min distances fine-grained approximation algorithm Theory of computation → Design and analysis of algorithms
Format(s):
Medium: X Size: 14 pages; 874462 bytes Other: application/pdf
Size(s):
14 pages 874462 bytes
Right(s):
Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
Sponsoring Org:
National Science Foundation
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