On the Online Weighted Non-Crossing Matching Problem

2026-03-10

AI Summary

This study addresses the online weighted non-crossing matching problem in the Euclidean plane, where points with non-negative weights arrive sequentially and must be immediately matched or left unmatched, with the goal of maximizing total weight under the constraint that matching edges do not intersect. By integrating online algorithm analysis, competitive ratio theory, randomized strategies, and geometric techniques for handling non-crossing constraints, the work establishes—for the first time—the existence of a randomized algorithm achieving a constant competitive ratio under arbitrary weights, thereby overcoming a fundamental limitation of deterministic algorithms which cannot attain any non-trivial bound. Additionally, the paper improves the upper bound on advice complexity and derives new upper and lower bounds for several variants, including bounded weights, matchings with revocation, and collinear point sets.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Algorithmic accountability and transparency on the web

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.

Problem

• online matching
• weighted matching
• non-crossing constraint
• competitive ratio
• Euclidean plane

Innovation

• online matching
• non-crossing constraint
• competitive ratio
• randomized algorithm
• advice complexity