Bounding the Optimal Performance of Online Randomized Primal-Dual Methods
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912763005108224 |
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| author | Xu, Pan |
| author_facet | Xu, Pan |
| contents | 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 (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 $F$. 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 (SICOMP 2024). Meanwhile, our analysis yields an upper bound of $0.5810$ for a function space strictly larger than that considered in Huang and Zhang (SICOMP 2024). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounding the Optimal Performance of Online Randomized Primal-Dual Methods Xu, Pan Data Structures and Algorithms 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 (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 $F$. 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 (SICOMP 2024). Meanwhile, our analysis yields an upper bound of $0.5810$ for a function space strictly larger than that considered in Huang and Zhang (SICOMP 2024). |
| title | Bounding the Optimal Performance of Online Randomized Primal-Dual Methods |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2503.09508 |