Learning-Augmented Online Bipartite Fractional Matching

TL;DR

This work develops learning-augmented algorithms for online bipartite fractional matching that leverage per-arrival fractional advice to improve robustness and consistency beyond the CoinFlip baseline. The LAB algorithm, tailored for vertex-weighted matching, and the PAW algorithm, designed for unweighted cases, provably dominate prior approaches and extend LAB to AdWords under small bids. A complementary hardness bound delineates limits on the robustness-consistency tradeoff, while experiments on synthetic and real data validate the practical efficacy and illuminate how advice quality impacts performance. Collectively, the results advance the design of prediction-informed online matching with provable guarantees and broad applicability to AdWords-like settings.

Abstract

Online bipartite matching is a fundamental problem in online optimization, extensively studied both in its integral and fractional forms due to its theoretical significance and practical applications, such as online advertising and resource allocation. Motivated by recent progress in learning-augmented algorithms, we study online bipartite fractional matching when the algorithm is given advice in the form of a suggested matching in each iteration. We develop algorithms for both the vertex-weighted and unweighted variants that provably dominate the naive "coin flip" strategy of randomly choosing between the advice-following and advice-free algorithms. Moreover, our algorithm for the vertex-weighted setting extends to the AdWords problem under the small bids assumption, yielding a significant improvement over the seminal work of Mahdian, Nazerzadeh, and Saberi (EC 2007, TALG 2012). Complementing our positive results, we establish a hardness bound on the robustness-consistency tradeoff that is attainable by any algorithm. We empirically validate our algorithms through experiments on synthetic and real-world data.

Figures (13)

• Figure 1: Robustness-consistency tradeoffs of previous works and our results.
• Figure 2: $f_0$, $f_1$, and $f$ with $\lambda \in \{0.1, 0.3, 0.5\}$. (a)-(c) depict the function values of $f_0$ and $f_1$ with respect to $z \in [0, 1]$. (d)-(f) depict the contour plots with respect to $A \in [0, 1]$ and $X \in [0, 1]$: the brighter the color is, the closer to 1 the function value is.
• Figure 3: Contour plots of $f(A, X)$ for $\lambda \in \{0.1, 0.3, 0.5\}$. We partition $[0,1]^2$ into three regions: $\mathcal{D}_\mathsf{L}$ (Left, in blue), $\mathcal{D}_\mathsf{BR}$ (Bottom Right, in green), $\mathcal{D}_\mathsf{TR}$ (Top Right, in yellow). The red curve represents $X = A - \ln A + 1 - \lambda + \ln \lambda$ on $A \in[\lambda, \lambda e^{1-\lambda}]$ that separates $\mathcal{D}_\mathsf{L}$ and $\mathcal{D}_\mathsf{TR}$.
• Figure 4: Illustration of the proof of \ref{['lem:vw:fr:reduce']} in the case where $(A_k, X_k) \in \mathcal{D}_\mathsf{BR}$.
• Figure 5: Illustration of the proof of \ref{['lem:vw:fr:reduce']} in the case where $(A_k, X_k) \in \mathcal{D}_\mathsf{L} \cup \mathcal{D}_\mathsf{TR}$.

Theorems & Definitions (83)

• Theorem 1
• Theorem 2
• Definition 4: Robustness and Consistency
• Lemma 5: see, e.g., devanur2013randomizedfahrbach2022edge
• proof
• Lemma 6
• Theorem 6
• Lemma 7
• proof
• proof