Hardness of Approximate Diameter: Now for Undirected Graphs

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 delta > 0, a 2 - k/1 - 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.