[Paper Review] Online Primal Dual Meets Online Matching with Stochastic Rewards: Configuration LP to the Rescue

Zhiyi Huang, Qiankun Zhang|arXiv (Cornell University)|Feb 5, 2020

Optimization and Search Problems28 references4 citations

TL;DR

This paper introduces a novel application of the randomized online primal dual framework to online matching with stochastic rewards by employing the configuration linear program instead of the standard matching LP. It achieves a competitive ratio of 0.572 for unequal, vanishing probabilities—surpassing the prior best bound of 0.534—and 0.576 for equal, vanishing probabilities, resolving a long-standing open problem in the field.

ABSTRACT

Mehta and Panigrahi (FOCS 2012) introduce the problem of online matching with stochastic rewards, where edges are associated with success probabilities and a match succeeds with the probability of the corresponding edge. It is one of the few online matching problems that have defied the randomized online primal dual framework by Devanur, Jain, and Kleinberg (SODA 2013) thus far. This paper unlocks the power of randomized online primal dual in online matching with stochastic rewards by employing the configuration linear program rather than the standard matching linear program used in previous works. Our main result is a 0.572 competitive algorithm for the case of vanishing and unequal probabilities, improving the best previous bound of 0.534 by Mehta, Waggoner, and Zadimoghaddam (SODA 2015) and, in fact, is even better than the best previous bound of 0.567 by Mehta and Panigrahi (FOCS 2012) for the more restricted case of vanishing and equal probabilities. For vanishing and equal probabilities, we get a better competitive ratio of 0.576. Our results further generalize to the vertex-weighted case due to the intrinsic robustness of the randomized online primal dual analysis.

Motivation & Objective

• To close the gap between known upper and lower bounds for online matching with stochastic rewards, particularly in the case of vanishing and unequal success probabilities.
• To extend the randomized online primal dual framework—previously ineffective on this problem—by leveraging the configuration linear program instead of the standard matching LP.
• To achieve improved competitive ratios in both the unequal and equal probability regimes, especially in the vanishing probability limit.
• To generalize results to the vertex-weighted setting through the robustness of the configuration LP and randomized primal dual analysis.

Proposed method

• Replaces the standard matching LP with the configuration LP to better model the stochastic matching process and amortize load across online vertices.
• Introduces a dual variable update rule based on thresholding and load balancing, where offline vertices are assigned thresholds to control matching behavior.
• Uses a gain-sharing mechanism that splits the expected reward $ p $ of each matched edge between the dual variables $ \alpha_u $ and $ \beta_v $ of its endpoints.
• Employs a differential inequality approach to derive the optimal dual function $ f(\theta) $, ensuring the expected dual gain meets the competitive ratio requirement.
• Analyzes the expected contribution of dual variables $ \alpha_u $ and $ \beta_v $ under random thresholds, with $ \beta_v $ dropping to zero when $ \theta_u < \ell_u^\infty $, a key bottleneck in standard LP analysis.
• Demonstrates that the configuration LP avoids the sharp drop in online-side contribution seen in the standard LP, enabling higher competitive ratios.

Experimental results

Research questions

• RQ1 Can the randomized online primal dual framework be successfully applied to online matching with stochastic rewards, a problem previously resistant to such methods?
• RQ2 Why does the standard matching LP fail to achieve competitive ratios above 0.5 in this setting, and what structural limitation does it expose?
• RQ3 Can the configuration LP mitigate the loss in dual contribution on the online side when thresholds fall below full load capacity?
• RQ4 What competitive ratio can be achieved using the configuration LP within the randomized online primal dual framework for unequal and vanishing success probabilities?
• RQ5 How does the competitive ratio compare to prior algorithms like Stochastic Balance and Semi-Adaptive in both equal and unequal probability regimes?

Key findings

• The proposed algorithm achieves a competitive ratio of 0.572 for online matching with stochastic rewards under unequal and vanishing success probabilities, improving upon the prior best bound of 0.534.
• For the case of equal and vanishing probabilities, the competitive ratio reaches 0.576, surpassing the previous best bound of 0.567 from Stochastic Balance.
• The configuration LP enables a graceful decay in online-side dual contribution, avoiding the abrupt drop to zero seen in standard LP analysis, which explains the improved performance.
• The method successfully extends to the vertex-weighted setting due to the robustness of the randomized online primal dual analysis under the configuration LP framework.
• The analysis proves that the standard matching LP cannot achieve better than 0.5 competitive ratio in this setting due to a fundamental bottleneck in dual contribution from the online side.
• The optimal dual function $ f(\theta) = 0.5 $ for all $ \theta \geq 0 $ yields a competitive ratio of 0.5 under the standard LP, confirming its limitation.