Bounding the Optimal Performance of Online Randomized Primal-Dual Methods

Pan Xu

Paper

Bounding the Optimal Performance of Online Randomized Primal-Dual Methods

Abstract

The online randomized primal-dual method has widespread applications in online algorithm design and analysis. A key challenge is identifying an appropriate function space,

, in which we search for an optimal updating function

that yields the best possible lower bound on the competitiveness of a given algorithm. The choice of

must balance two competing objectives: on one hand, it should impose sufficient simplifying conditions on

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

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 (SICOMP 2024), 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

. Notably, we establish that Stochastic Balance achieves a competitiveness of at least

for the problem (under equal vanishing probabilities), improving upon the previous bound of

by Huang and Zhang (SICOMP 2024). Meanwhile, our analysis yields an upper bound of

for a function space strictly larger than that considered in Huang and Zhang (SICOMP 2024).