We study the problem of approximating the diameter of directed graphs under the min-distance measure, defined as $d_{\min}(u,v) = \min(d(u,v), d(v,u))$. Unlike standard shortest-path distance, min-distance is not a metric, which renders many classical techniques inapplicable. Prior work has therefore focused on approximating this parameter, culminating in an approximation-runtime tradeoff by Dalirrooyfard et al. [ICALP'19] giving a $4k-1$ approximation in $\tilde{O}(mn^{1/(k+1)})$ time for any positive integer $k$ and, more recently, the first near-linear time constant approximation by Chechik and Zhang [FOCS'22], where they obtained a 4-approximation to the min-diameter. In this work we present a randomized near-linear time algorithm that achieves a $3$-approximation to the min-diameter, outperforming all known approximation-runtime tradeoffs. Our approach introduces a novel type-classification framework that may be of independent interest. We further extend our techniques to the more general setting of multimode graphs, recently introduced as a generalization of min-distance by Kirkpatrick and Vassilevska W. [MFCS'25]. For directed $2$-mode graphs, we obtain a $3$-approximation to the diameter in near-linear time, dramatically improving over the previously best known $n$-approximation. Our results significantly narrow the gap between min-distance and multimode distance approximations, and open new directions for understanding graph parameters under non-metric distance measures.
The sensitivity setting is a restricted setting for dynamic algorithms, particularly practical for scenarios where extensive preprocessing is feasible but responses to real-time modifications must be near-instantaneous before the data structure is eventually rebuilt. For graph problems, a sensitivity data structure is constructed with a preprocessing time P so that the following queries can be answered quickly, preferably in $O(1)$ time: given an edge $e$, return the answer to the problem on either $G \setminus e$ (decremental) or $G \cup e$ (incremental). In this paper, we almost entirely settle the decremental setting for the diameter and eccentricities problems in a variety of approximation regimes by matching P to the static runtime while supporting $O(1)$-time queries, thereby improving upon all previous results for a single failure [Bil\`o, Cohen, Friedrich, Schirneck, MFCS 2021; Bil\`o, Choudhary, Cohen, Friedrich, Krogmann, Schirneck, ICALP 2021]. More precisely: (1) We provide a tight reduction demonstrating that any exact distance sensitivity oracle can be used to efficiently solve decremental exact diameter and all-node eccentricities; (2) For the approximate setting, we match the runtime of all known static diameter algorithms across all sparsity settings, up to an additional $1+o(1)$ factor in approximation. Conversely, for the previously unexplored incremental setting of these problems: (3) We develop new lower bounds, demonstrating that no incremental algorithm can efficiently approximate diameter, radius, or eccentricity beyond a $5/3$ factor in undirected graphs or a $2$ factor in directed graphs; (4) We introduce two new instructive techniques and demonstrate how to utilize them to construct several new algorithms. Most notably, we develop incremental single-node eccentricity approximations for both directed and undirected graphs that match our new lower bounds.
Sam Hiken, Yael Kirkpatrick, Jakob Nogler et al.· 0 citations