Computer Science > Computational Complexity
[Submitted on 10 Jun 2021 (v1), last revised 15 Nov 2021 (this version, v2)]
Title:Hardness of Approximate Diameter: Now for Undirected Graphs
View PDFAbstract: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 $k\ge 2$ and $\delta>0$, a $2-\frac{1}{k}-\delta$ approximation for diameter in directed $m$-edge graphs requires $mn^{1+1/(k-1)-o(1)}$ 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'13], [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 develop new tools for fine-grained reductions that could be useful for proving SETH-based hardness for other problems in undirected graphs related to distance computation.
Submission history
From: Mina Dalirrooyfard [view email][v1] Thu, 10 Jun 2021 20:07:24 UTC (653 KB)
[v2] Mon, 15 Nov 2021 17:24:04 UTC (1,365 KB)
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
Connected Papers (What is Connected Papers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.