Optimal Competitive Ratio of Two-sided Online Bipartite Matching

We establish an optimal upper bound (negative result) of $\sim 0.526$ on the competitive ratio of the fractional version of online bipartite matching with two-sided vertex arrivals, matching the lower bound (positive result) achieved by Wang and Wong (ICALP 2015), and Tang and Zhang (EC 2024).

💡 Research Summary

The paper addresses the two‑sided online bipartite matching problem, where vertices from both sides of a bipartite graph arrive one by one in an adversarial order. Upon the arrival of a vertex, all incident edges to previously seen vertices are revealed, and the algorithm must either match the new vertex immediately or defer it for future consideration. The objective is to maximize the size of the final matching, and performance is measured by the competitive ratio Γ, i.e., the worst‑case guarantee that the algorithm’s matching size is at least a Γ‑fraction of the offline optimum.

Historically, the one‑sided model (only one side known in advance) enjoys a tight competitive ratio of 1‑1/e ≈ 0.632, but the two‑sided model is provably harder. Prior work by Wang and Wong (ICALP 2015) and later by Tang and Zhang (EC 2024) presented fractional algorithms achieving a competitive ratio of approximately 0.526. On the impossibility side, a series of papers gradually raised the upper bound from 0.625 down to 0.584, leaving a gap between 0.526 and 0.584.

The main contribution of this work is to close that gap by proving a tight upper bound of ≈ 0.526 for the fractional version, thereby confirming that the known algorithms are optimal. The proof proceeds in two major phases.

Phase 1 – Recursive Function Construction.
The authors define a family of functions {Fₙ}ₙ≥1 on the interval