Bounding the Optimal Performance of Online Randomized Primal-Dual Methods

The online randomized primal-dual method has widespread applications in online algorithm design and analysis. A key challenge is identifying an appropriate function space, $F$, in which we search for an optimal updating function $f \in F$ that yields the best possible lower bound on the competitiveness of a given algorithm. The choice of $F$ must balance two competing objectives: on one hand, it should impose sufficient simplifying conditions on $f$ to facilitate worst-case analysis and establish a valid lower bound; on the other hand, it should remain general enough to offer a broad selection of candidate functions. The tradeoff is that any additional constraints on $f$ that can facilitate competitive analysis may also lead to a suboptimal choice, weakening the resulting lower bound. To address this challenge, we propose an auxiliary-LP-based framework capable of effectively approximating the best possible competitiveness achievable when applying the randomized primal-dual method to different function spaces. Specifically, we examine the framework introduced by Huang and Zhang (STOC 2020), which analyzes Stochastic Balance for vertex-weighted online matching with stochastic rewards. Our approach yields both lower and upper bounds on the best possible competitiveness attainable using the randomized primal-dual method for different choices of ${F}$. Notably, we establish that Stochastic Balance achieves a competitiveness of at least $0.5796$ for the problem (under equal vanishing probabilities), improving upon the previous bound of $0.576$ by Huang and Zhang (STOC 2020). Meanwhile, our analysis yields an upper bound of $0.5810$ for a function space strictly larger than that considered in Huang and Zhang (STOC 2020).