{"ID":6497592,"CreatedAt":"2026-07-13T01:19:40.13847098Z","UpdatedAt":"2026-07-14T01:36:59.12045529Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.09588","arxiv_id":"2607.09588","title":"Improved Approximation of Min-Distances in Near-Linear Time","abstract":"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.","short_abstract":"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 approxi...","url_abs":"https://arxiv.org/abs/2607.09588","url_pdf":"https://arxiv.org/pdf/2607.09588v1","authors":"[\"Yael Kirkpatrick\"]","published":"2026-07-10T16:39:18Z","proceeding":"cs.DS","tasks":"[\"cs.DS\"]","methods":"[]","has_code":false}