发表机构
Technische Universität Berlin; University of Bremen（柏林工业大学; 不来梅大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对多阶段在线二分匹配，提出首个任意阶段数的一致性-鲁棒性权衡算法，并扩展到在线设置和AdWords，通过带惩罚的凸规划实现最优或改进的权衡。
AI 中文摘要
我们研究了具有多个阶段的学习增强型在线二分匹配问题。在$k$阶段顶点加权分数二分匹配问题中，需求顶点分$k$个阶段到达，算法在每个阶段接收可能不准确的分配预测。虽然两阶段情形已知有紧的一致性与鲁棒性权衡，但对于任意阶段数此前没有非平凡权衡结果。我们的主要结果是针对带预测的$k$阶段顶点加权分数二分匹配的第一个一致性-鲁棒性权衡，适用于每个$k\ge2$。令$R_k=1-(1-1/k)^k$。对于每个$R\in[0,R_k]$，我们的算法是$R$-鲁棒的且$C_k(R)$-一致的，其中$C_k(R)=k(1-R)^{1/k}+R-(k-1)$。这同时恢复了已知的紧两阶段权衡和最优的无预测$k$阶段竞争保证$R_k = C_k(R_k)$，同时严格优于这些端点之间的自然随机抛硬币基线。我们还提出了经典在线设置下的算法，其中需求逐个到达且需求数量事先未知。对于给定的鲁棒性$R\in[0,1-1/e]$，它的一致性比率至少为$C_\infty(R)=1+R+\ln(1-R)$，改进了该问题先前已知的最佳权衡。最后，我们将框架扩展到分数AdWords和分数预测。我们的算法基于逐阶段凸规划，并带有精心校准的顶点相关惩罚。这些惩罚为每个供应顶点维持动态安全储备，平衡对抗未来对抗性到达的保护与利用预测分配的能力。
英文摘要
We study learning-augmented online bipartite allocation with multiple stages. In the $k$-stage vertex-weighted fractional bipartite matching problem, demand vertices arrive in $k$ stages, and the algorithm receives possibly inaccurate predictions of the allocation in each stage. While tight consistency-robustness tradeoffs were known for the two-stage case, no nontrivial tradeoff was known for an arbitrary number of stages. Our main result is the first consistency-robustness tradeoff for $k$-stage vertex-weighted fractional bipartite matching with predictions, for every $k\ge2$. Let $R_k=1-(1-1/k)^k$. For every $R\in[0,R_k]$, our algorithm is $R$-robust and $C_k(R)$-consistent, where $C_k(R)=k(1-R)^{1/k}+R-(k-1)$. This simultaneously recovers the known tight two-stage tradeoff and the optimal prediction-free $k$-stage competitive guarantee $R_k = C_k(R_k)$, while strictly dominating the natural randomized coin-flip baseline between these endpoints. We also present an algorithm for the classical online setting, where demands arrive one by one and the number of demands is unknown in advance. It has a consistency ratio of at least $C_\infty(R)=1+R+\ln(1-R)$ for a given robustness $R\in[0,1-1/e]$, improving the best previously known tradeoff for this problem. Finally, we extend the framework to fractional AdWords and fractional predictions. Our algorithms are based on stage-wise convex programs with carefully calibrated vertex-dependent penalties. The penalties maintain a dynamic safety reserve for each supply vertex, balancing protection against adversarial future arrivals with the ability to exploit the predicted allocation.
