Year: 2024
Type: article
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 articles 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 δ > 0, a \(2-\frac{1}{k}-\delta\) approximation for diameter in directed m -edge graphs requires \(m^{1+1/(k-1)-o(1)}\) time. In particular, the simple linear time 2-approximation algorithm is optimal for directed graphs. In this article, 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 conditionally 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.
Source: Journal of the ACM
DOI: 10.1145/3704631
Institutions: Santa Clara University, Massachusetts Institute of Technology
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: bronze
Funder: National Science Foundation
Awards: AF: Small: Average-Case Fine-Grained Complexity, Graduate Research Fellowship Program (GRFP), CAREER:Matrix Products: Algorithms and Applications, G6671297155, AF: Small: Graphs and structures for distance estimation, AF:Small: Algorithms and Limitations for Matrix Multiplication, AF: Medium: Collaborative Research: Hardness in Polynomial Time
OpenAlex ID: W4404417988