Hardness of Approximate Diameter: Now for Undirected Graphs
Name
3704631.pdf
Size
551.11 KB
Format
Adobe PDF
Checksum (MD5)
1fce5448e7586f626926fda173cf8670
Author(s)
Dalirrooyfard, Mina
Li, Ray
Vassilevska Williams, Virginia
Journal
Journal of the ACM
Publisher
ACM
Citation
Dalirrooyfard, Mina, Li, Ray and Vassilevska Williams, Virginia. "Hardness of Approximate Diameter: Now for Undirected Graphs." Journal of the ACM.
Version
Final published version
Abstract
Approximating the graph diameter is a basic task of both theoretical and practical interest. A simple folklore algorithm can output a 2-approximation to the diameter in linear time by running BFS from an arbitrary vertex. It has been open whether a better approximation is possible in near-linear time. A series of papers on fine-grained complexity have led to strong hardness results for diameter in directed graphs, culminating in a recent tradeoff curve independently discovered by [Li, STOC'21] and [Dalirrooyfard and Wein, STOC'21], showing that under the Strong Exponential Time Hypothesis (SETH), for any integer k?2 and ?>0, a (2-(1/k)-?) approximation for diameter in directed m-edge graphs requires mn1+1/(k-1)-o(1) time. In particular, the simple linear time 2-approximation algorithm is optimal for directed graphs. In this paper we prove that the same tradeoff lower bound curve is possible for undirected graphs as well, extending results of [Roditty and Vassilevska W., STOC', [Li'20] and [Bonnet, ICALP'21] who proved the first few cases of the curve, k=2,3 and 4, respectively. Our result shows in particular that the simple linear time 2-approximation algorithm is also optimal for undirected graphs. To obtain our result, we extract the core ideas in known reductions and introduce a unification and generalization that could be useful for proving SETH-based hardness for other problems in undirected graphs related to distance computation.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1145/3704631
