A randomized algorithm for the on-line weighted bipartite matching problem

We study the on-line minimum weighted bipartite matching problem in arbitrary metric spaces. Here, $n$ not necessary disjoint points of a metric space $M$ are given, and are to be matched on-line with $n$ points of $M$ revealed one by one. The cost of a matching is the sum of the distances of the matched points, and the goal is to find or approximate its minimum. The competitive ratio of the deterministic problem is known to be $\Theta(n)$. It was conjectured that a randomized algorithm may perform better against an oblivious adversary, namely with an expected competitive ratio $\Theta(\log n)$. We prove a slightly weaker result by showing a $o(\log^3 n)$ upper bound on the expected competitive ratio. As an application the same upper bound holds for the notoriously hard fire station problem, where $M$ is the real line.

💡 Research Summary

The paper addresses the online minimum‑weight bipartite matching problem in an arbitrary metric space M. We are given a fixed set X of n points (the “servers”) in M, and a sequence of n request points Y that arrive one by one. When a request yₖ appears, the algorithm must immediately match it to an unmatched point xᵢ∈X; the total cost is the sum of the distances d(xᵢ, yₖ). In the deterministic setting the optimal competitive ratio is Θ(n), meaning that in the worst case an online algorithm can be a linear factor worse than the offline optimum.

The authors focus on a randomized algorithm against an oblivious adversary (the adversary fixes the entire request sequence in advance and does not see the algorithm’s random choices). Their goal is to improve the competitive ratio from linear to polylogarithmic. They present a novel algorithm whose expected competitive ratio is o(log³ n), a substantial improvement over the deterministic bound and a step toward the conjectured Θ(log n) bound.

The core technical contribution is a hierarchical random partition of the metric space. For each level i = 0,…,L (where L = ⌈log₂ Δ⌉ and Δ is the diameter of M) the algorithm draws a random shift ξᵢ∈