Near-Optimal Online Metric Matching on $Δ$-ary HST

2026-09-21

AI Summary

本文针对在线度量匹配问题，通过在每个节点最多有Δ个子节点的HST上设计算法，实现了与n无关、接近最优的竞争比O((log log Δ)·log Δ)。

Abstract

In the online metric matching problem, we have $n$ servers with known locations in some metric space. Requests arrive one-by-one at certain locations, and upon arrival a request must be matched to a server that was not matched to a previous request. The goal is to minimize the matching cost. For randomized algorithms with an oblivious adversary, the best known competitive ratio is obtained by embedding the metric space into an HST, and then solving the problem in the setting where the metric space is defined by the HST. Bansal et al. (Algorithmica, 2014) introduced a framework for online metric matching where one develops an algorithm in a restricted reassignment model, and then transforms this into a true online algorithm. Using this framework, they obtained an expected competitive ratio of $O(\log n)$ for HSTs; this also gives the best known competitive ratio of $O(\log^2 n)$ for general metrics. In this paper, we revisit this framework with the aim of developing new algorithms. For HSTs where each node has at most $Δ$ children, we develop an algorithm via this framework with an expected competitive ratio of $O((\log\log Δ) \cdot \log Δ)$. In particular, this ratio is independent of $n$, the number of servers/requests. It is near-optimal, as the expected competitive ratio of any algorithm is $Ω(\log Δ)$.

Problem

• online metric matching
• HST
• competitive ratio

Innovation

• Online Metric Matching
• HST
• Competitive Ratio
• $\Delta$-ary HST
• Reassignment Model