On the Online Weighted Non-Crossing Matching Problem

TL;DR

This paper investigates the online weighted non-crossing matching problem in the Euclidean plane, demonstrating that randomized algorithms can achieve constant competitive ratios, unlike deterministic ones, and explores various problem variants and advice complexity bounds.

Contribution

It introduces the weighted online non-crossing matching problem, provides bounds for deterministic and randomized algorithms, and analyzes problem variants and advice complexity.

Findings

Deterministic algorithms cannot guarantee a non-trivial competitive ratio.

Randomized algorithms can achieve a constant competitive ratio for arbitrary weights.

Bounds are established for problem variants like revocability and collinearity.

Abstract

We introduce and study the weighted version of an online matching problem in the Euclidean plane with non-crossing constraints: 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 classic 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, but we give upper and lower bounds on the problem with bounded weights. In contrast to the deterministic case, we show that using randomization, a constant competitive ratio is possible for arbitrary weights. We also study other variants of the problem, including revocability and collinear points, both of which permit non-trivial online algorithms, and we give upper and lower bounds for the attainable competitive ratios. Finally, we prove an advice complexity bound for obtaining optimality, improving the best known bound.