Online bipartite matching with imperfect advice

TL;DR

This paper investigates online bipartite matching, demonstrating limitations of learning-augmented algorithms under adversarial models and proposing advice-based algorithms that improve performance under random arrivals.

Contribution

It proves that no learning-augmented method can surpass 1/2-robustness while being 1-consistent in adversarial settings, and introduces advice-based algorithms that interpolate between known bounds under random arrivals.

Findings

No learning-augmented method can be both 1-consistent and better than 1/2-robust in adversarial models.

Advice-based algorithms can achieve a spectrum of competitive ratios under random arrival models.

The proposed algorithms adaptively improve performance based on advice quality.

Abstract

We study the problem of online unweighted bipartite matching with $n$ offline vertices and $n$ online vertices where one wishes to be competitive against the optimal offline algorithm. While the classic RANKING algorithm of Karp et al. [1990] provably attains competitive ratio of $1-1/e > 1/2$, we show that no learning-augmented method can be both 1-consistent and strictly better than $1/2$-robust under the adversarial arrival model. Meanwhile, under the random arrival model, we show how one can utilize methods from distribution testing to design an algorithm that takes in external advice about the online vertices and provably achieves competitive ratio interpolating between any ratio attainable by advice-free methods and the optimal ratio of 1, depending on the advice quality.