Consistency-Robustness Tradeoffs for Online Bipartite Allocation with Multiple Stages

RESUMO

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.