arXiv:2611.17090 · v1

Online Transportation via Shifted Geometric Scales

首次提交 2026-11-13 00:53

版本更新 2026-11-13 00:53

雷达观测 2026-11-13 14:15

摘要

In the Online Transportation problem, introduced by Kalyanasundaram and Pruhs (1995), requests arrive sequentially in a metric space and must be assigned immediately and irrevocably to one of $m$ facilities with available capacity. The objective is to minimize the total distance between requests and their assigned facilities. The problem generalizes the Online Metric Matching problem, in which each facility has unit capacity, and inherits a $2m-1$ lower bound on the competitive ratio for all deterministic algorithms. For Online Metric Matching, Kalyanasundaram and Pruhs (1993) and Khuller, Mitchell, and Vazirani (1994) established an optimal deterministic competitive ratio of $2m-1$, attained by the Retrospective (Permutation) algorithm. Kalyanasundaram and Pruhs (1998) conjectured that the same ratio is achievable for online transportation. Recently, Seth, Das, and Raghvendra (2026) obtained a competitive ratio of approximately $6.66m-2.89$ using their Robustified Greedy algorithm. We give an algorithm with competitive ratio $ρ(m-1)+1$, where $ρ\approx5.64$. Our algorithm builds on minimum-spanning-tree search techniques developed by Harada and Itoh (2025) and Arndt, Moseley, Pruhs, and Uetz (2026), using priority-based virtual depth-first searches over the known facilities. However our algorithm uses a phase to shift the geometric search scales, to decide when to skip certain searches and optimize the thresholds that define what edges to include in a given search.

作者备注：The majority of this work was generated and edited by iteratively prompting ChatGPT-6 Astra. After review, we believe the results to be correct and want to post the work for the community to view, but we are actively working on simplifying the prose and proof to more clearly highlight the technical insights and contributions