On the Online Weighted Non-Crossing Matching Problem.

Joan Boyar, Shahin Kamali, Kim S. Larsen, Ali Mohammad Lavasani, Yaqiao Li, and Denis Pankratov.
In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), volume 294 of Leibniz International Proceedings in Informatics (LIPIcs), pages 16:1-16:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH, 2024.

We introduce and study the weighted version of an online matching problem in the Euclidean plane with non-crossing constraints: \( 2n \) points with non-negative weights arrive online, and an algorithm can match an arriving point to one of the unmatched previously arrived points. In the vanilla model, the decision on how to match (if at all) a newly arriving point is irrevocable. The goal is to maximize the total weight of matched points under the constraint that straight-line segments corresponding to the edges of the matching do not intersect. The unweighted version of the problem was introduced in the offline setting by Atallah in 1985, and this problem became a subject of study in the online setting with and without advice in several recent papers.

We observe that deterministic online algorithms cannot guarantee a non-trivial competitive ratio for the weighted problem. We study various regimes of the problem which permit non-trivial online algorithms. In particular, when weights are restricted to the interval \( [1, U] \) we give a deterministic algorithm achieving competitive ratio \( \Omega\left(2^{-2\sqrt{\log U}}\right) \). We also prove that deterministic online algorithms cannot achieve competitive ratio better than \( O\left(2^{-\sqrt{\log U}}\right) \). Interestingly, we establish that randomization alone suffices to achieve competitive ratio \( 1/3 \) even when there are no restrictions on the weights. Additionally, if one allows an online algorithm to revoke acceptances, then one can achieve a competitive ratio \( \approx 0.2862 \) deterministically for arbitrary weights. We also establish a lower bound on the competitive ratio of randomized algorithms in the unweighted setting, and improve the best-known bound on advice complexity to achieve a perfect matching.

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