Online minimum matching with uniform metric and random arrivals

TL;DR

This paper analyzes the online minimum bipartite matching problem under uniform metric with random arrivals, showing that a randomized greedy algorithm achieves a near-optimal competitive ratio, similar to the adversarial case.

Contribution

It establishes that the randomized greedy algorithm's performance under random arrivals matches the lower bound, indicating no advantage over adversarial order.

Findings

RG achieves a competitive ratio of (1+1/n)(H_{n+1}-1)

Random order does not improve RG's performance over adversarial order

The model's complexity remains similar under random arrivals and adversarial order

Abstract

We consider Online Minimum Bipartite Matching under the uniform metric. We show that Randomized Greedy achieves a competitive ratio equal to $(1+1/n) (H_{n+1}-1)$, which matches the lower bound. Comparing with the fact that RG achieves an optimal ratio of $\Theta(\ln n)$ for the same problem but under the adversarial order, we find that the weaker arrival assumption of random order doesn't offer any extra algorithmic advantage for RG, or make the model strictly more tractable.