Optimal Competitive Ratio of Two-sided Online Bipartite Matching

TL;DR

This work resolves the optimal competitive ratio for the fractional two-sided online bipartite matching problem. It pairs a Tang–Zhang style analysis with a new recursive adversarial construction that yields a sequence of functions $\{F_n\}$ showing that any $\Gamma>\Gamma^*$ cannot be achieved; the limit constant is $\Gamma^* = \max_{k\ge1} \frac{1}{(\frac{k+1}{2})^{\frac{k+1}{2k}}(\frac{k-1}{2})^{\frac{k-1}{2k}}+1} \approx 0.526$. The proof combines meticulous mathematical facts about these functions, a base and recursive instance construction, and a two-branch analysis against aggressive and conservative algorithms to bound the achievable performance. Consequently, the room for improvement over the known $0.526$-competitive fractional algorithms is essentially exhausted, and the findings highlight the intrinsic difficulty of two-sided arrivals relative to fully online models.

Abstract

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).

Theorems & Definitions (11)

• Theorem 1.1
• Claim 3.1
• Claim 3.2
• Claim 3.3
• Lemma 3.1
• Lemma 3.2
• Claim 3.4
• Claim 4.1
• Claim 4.2
• Claim 4.3