Learning-Augmented Online Bipartite Matching in the Random Arrival Order Model

TL;DR

The paper tackles online unweighted bipartite matching in the random arrival order model with untrusted type predictions for online vertices. It generalizes prior learning-augmented approaches by removing the assumption that the optimum matches all online vertices and only requires the predicted matching size to be at least αn, achieving $(1-o(1))$-consistency and $(β-o(1))$-robustness (with β ≈ 0.696) and showing a smooth degradation of performance as prediction error grows. The authors introduce Test-and-Match+, a sampling-based algorithm that estimates the L1 distance between the true and predicted type distributions and switches between a prediction-guided Mimic strategy and the Baseline online algorithm based on a threshold; they provide tight bounds on the number of remaining optimal matches and the probability of sampling failures. The results substantially improve robustness for imperfect predictions and demonstrate graceful trade-offs between consistency and robustness, with potential extensions to sublinear prediction regimes.

Abstract

We study the online unweighted bipartite matching problem in the random arrival order model, with $n$ offline and $n$ online vertices, in the learning-augmented setting: The algorithm is provided with untrusted predictions of the types (neighborhoods) of the online vertices. We build upon the work of Choo et al. (ICML 2024, pp. 8762-8781) who proposed an approach that uses a prefix of the arrival sequence as a sample to determine whether the predictions are close to the true arrival sequence and then either follows the predictions or uses a known baseline algorithm that ignores the predictions and is $β$-competitive. Their analysis is limited to the case that the optimal matching has size $n$, i.e., every online vertex can be matched. We generalize their approach and analysis by removing any assumptions on the size of the optimal matching while only requiring that the size of the predicted matching is at least $αn$ for any constant $0 < α\le 1$. Our learning-augmented algorithm achieves $(1-o(1))$-consistency and $(β-o(1))$-robustness. Additionally, we show that the competitive ratio degrades smoothly between consistency and robustness with increasing prediction error.

Figures (1)

• Figure 1: Illustration of vertex types

Theorems & Definitions (10)

• Theorem 1: Choo et al. online-bipartite-matching-imperfect-advice
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5
• Corollary 6
• Lemma 7
• Corollary 8
• Lemma 9
• Lemma 10