Revisiting Ranking for Online Bipartite Matching with Random Arrivals: the Primal-Dual Analysis

TL;DR

The paper analyzes Ranking for online bipartite matching under random arrivals through a randomized primal-dual lens, establishing that the best competitive ratio achievable within this framework lies in the interval $[0.6862, 0.7027]$ by solving discretized variational formulations via LPs. It introduces an independent random arrival model to smoothly interpolate between worst-case and random arrivals, demonstrating improved lower bounds (e.g., $0.6656$ for $m=2$ and $0.6763$ for $m=3$) and, overall, confirming that the primal-dual approach can approach the classical $0.696$ benchmark. The methodology relies on discretizing the $f$-function, representing $\alpha$ and $\beta$ as grid paths, and deriving two linear programs that bound the core max-min optimization from below and above, with numerical results aligning with and extending prior insights (Mahdian and Yan; Jin and Williamson). The findings bridge the unweighted and vertex-weighted settings in the random arrival model and offer computationally guided estimates that inch toward a tighter understanding of Ranking’s performance in this regime, while leaving open questions about closing the remaining gap and achieving tighter than $0.703$ bounds in general. The work thereby provides a practical pathway to near-optimal analysis via discretization and LP-based verification, with implications for related stochastic and online matching problems.

Abstract

We revisit the celebrated Ranking algorithm by Karp, Vazirani, and Vazirani (STOC 1990) for online bipartite matching under the random arrival model, that is shown to be $0.696$-competitive for unweighted graphs by Mahdian and Yan (STOC 2011) and $0.662$-competitive for vertex-weighted graphs by Jin and Williamson (WINE 2021). In this work, we explore the limitation of the primal-dual analysis of Ranking and aim to bridge the gap between unweighted and vertex-weighted graphs. We show that the competitive ratio of Ranking is between $0.686$ and $0.703$, under our current knowledge of Ranking and the framework of primal-dual analysis. This confirms a conjecture by Huang, Tang, Wu, and Zhang (TALG 2019), stating that the primal-dual analysis could lead to a competitive ratio that is very close to $0.696$. Our analysis involves proper discretizations of a variational problem and uses LP solver to pin down the numerical number. As a bonus of our discretization approach, our competitive analysis of Ranking applies to a more relaxed random arrival model. E.g., we show that even when each online vertex arrives independently at an early or late stage, the Ranking algorithm is at least $0.665$-competitive, beating the $1-1/e \approx 0.632$ competitive ratio under the adversarial arrival model.

Figures (5)

• Figure 1: Illustration of Lemma \ref{['lem:alpha_beta']} and \ref{['lem:optimization']}
• Figure 2: Illustration of the discretization for $m=3,n=4$
• Figure 3: The red vertices correspond to the online vertices and the numbers written in the circles correspond to their arrival time. The blue vertices correspond to the offline vertices and the number written in the circles correspond to their ranks.
• Figure 4: Local Perturbations
• Figure 5: Local search algorithm for $n=20$

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2: Informal, refer to Theorem \ref{['thm:opt']}
• Theorem 1.3
• Lemma 2.1: Lemma 3.1 of Huang et al. talg/HuangTWZ19. Refer to Figure \ref{['fig:matching_status']}
• Lemma 2.2
• proof
• Remark 2.1
• Theorem 2.1
• Definition 3.1
• Lemma 3.1