[Paper Review] Online Matching with Stochastic Rewards: Advanced Analyses Using Configuration Linear Programs

Zhiyi Huang, Hanrui Jiang+4 more|arXiv (Cornell University)|Sep 19, 2023

Optimization and Search ProblemsComputer Science3 citations

TL;DR

This paper advances online matching with stochastic rewards using Configuration Linear Programs (LPs), introducing a novel Stochastic Configuration LP that simplifies the Path-based LP of Goyal and Udwani (2020). It achieves improved competitive ratios: 0.611 for Stochastic Balance under infinitesimal success probabilities and 0.613 when probabilities are equal, while also providing the first online primal-dual analysis of the Ranking algorithm via Configuration LP, boosting its competitive ratio from 0.534 to 0.572.

ABSTRACT

Mehta and Panigrahi (2012) proposed Online Matching with Stochastic Rewards, which generalizes the Online Bipartite Matching problem of Karp, Vazirani, and Vazirani (1990) by associating the edges with success probabilities. This new feature captures the pay-per-click model in online advertising. Recently, Huang and Zhang (2020) studied this problem under the online primal dual framework using the Configuration Linear Program (LP), and got the best known competitive ratios of the Stochastic Balance algorithm. Their work suggests that the more expressive Configuration LP is more suitable for this problem than the Matching LP. This paper advances the theory of Configuration LP in two directions. Our technical contribution includes a characterization of the joint matching outcome of an offline vertex and \emph{all its neighbors}. This characterization may be of independent interest, and is aligned with the spirit of Configuration LP. By contrast, previous analyses of Ranking generally focus on only one neighbor. Second, we designed a Stochastic Configuration LP that captures a stochastic benchmark proposed by Goyal and Udwani (2020), who used a Path-based LP. The Stochastic Configuration LP is smaller and simpler than the Path-based LP. Moreover, using the new LP we improved the competitive ratio of Stochastic Balance from $0.596$ to $0.611$ when the success probabilities are infinitesimal, and to $0.613$ when the success probabilities are further equal.

Motivation & Objective

• To address the limitations of Matching LP in analyzing online matching with stochastic rewards, particularly in capturing joint matching outcomes of offline vertices and their neighbors.
• To develop a more expressive and simplified LP formulation—Stochastic Configuration LP—that captures the stochastic benchmark proposed by Goyal and Udwani (2020).
• To enable the first online primal-dual analysis of the Ranking algorithm using Configuration LP, which previously failed under standard frameworks.
• To improve competitive ratios for Stochastic Balance and Ranking by leveraging advanced LP-based analysis techniques.
• To demonstrate the superiority of Configuration LP over Matching LP and Path-based LP in expressive power and analytical tractability for stochastic online matching.

Proposed method

• Introduce a new characterization of the joint matching outcome of an offline vertex and all its neighbors, enabling more expressive dual variable analysis.
• Design a Stochastic Configuration LP that captures the stochastic benchmark, replacing the more complex Path-based LP with a smaller, simpler formulation.
• Apply the online primal-dual framework to the Stochastic Configuration LP, using a carefully designed gain-splitting rule to satisfy dual feasibility and competitive ratio constraints.
• Derive differential inequalities involving the gain-splitting function $ g $, and optimize $ g $ to maximize the competitive ratio under worst-case conditions.
• Use the function $ g(x) = e^{x-1} $ in the analysis of Ranking, which ensures the dual variables satisfy the required inequality with equality for the $ 1 - 1/e $ competitive ratio.
• Condition the analysis on the ranks and stochastic thresholds of other vertices, and apply lower bounds on expected dual variables to derive the competitive ratio.

Experimental results

Research questions

• RQ1 Can the Configuration LP be used to provide a non-trivial competitive analysis for the Ranking algorithm in online matching with stochastic rewards?
• RQ2 Can a simplified and more expressive LP formulation replace the Path-based LP of Goyal and Udwani (2020) while preserving or improving competitive ratios?
• RQ3 What is the optimal gain-splitting rule in the online primal-dual framework when using the Configuration LP for stochastic matching?
• RQ4 How does the joint matching behavior of an offline vertex and its neighbors influence the competitive ratio in stochastic online matching?
• RQ5 Can the competitive ratio of Stochastic Balance be improved beyond 0.596 using Configuration LP-based analysis?

Key findings

• The paper achieves a competitive ratio of 0.611 for the Stochastic Balance algorithm under infinitesimal success probabilities, improving upon the prior bound of 0.596.
• When success probabilities are equal, the competitive ratio for Stochastic Balance is improved to 0.613 using the new Stochastic Configuration LP.
• The first online primal-dual analysis of the Ranking algorithm using Configuration LP is established, achieving a competitive ratio of 0.572, up from the prior 0.534.
• The Stochastic Configuration LP is shown to be smaller and simpler than the Path-based LP of Goyal and Udwani (2020), while capturing the same stochastic benchmark.
• The analysis confirms that the dual variables derived from the Configuration LP satisfy the required inequalities, ensuring competitive ratio guarantees under the new formulation.
• The choice $ g(x) = e^{x-1} $ ensures that the dual expectation bound achieves the $ 1 - 1/e $ competitive ratio for the stochastic benchmark, validating the theoretical framework.