Consistency-Robustness Tradeoffs for Online Bipartite Allocation with Multiple Stages

2026-09-14

AI Summary

本文研究了多阶段在线二分图分配问题，通过阶段性的凸规划和顶点依赖惩罚方法，首次为k阶段问题提供了鲁棒性和一致性之间的权衡解。

Abstract

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.

Problem

• online bipartite allocation
• multiple stages
• consistency-robustness tradeoffs
• vertex-weighted fractional bipartite matching

Innovation

• k-stage vertex-weighted fractional bipartite matching
• consistency-robustness tradeoff
• stage-wise convex programs
• vertex-dependent penalties
• dynamic safety reserve