[论文解读] Attenuate Locally, Win Globally: An Attenuation-based Framework for Online Stochastic Matching with Timeouts

Brian Brubach, Karthik Abinav Sankararaman+2 more|arXiv (Cornell University)|Apr 22, 2018

Optimization and Search Problems参考文献 24被引用 3

一句话总结

本文提出了一种基于衰减的在线框架，用于处理带有超时限制的随机匹配问题，将离线随机匹配算法作为黑箱使用。该框架在标准问题中实现了0.46的竞争比，在双侧超时问题中首次实现了0.30的竞争比，显著优于以往结果，同时通过更优的离线黑箱算法为未来改进提供了可能。

ABSTRACT

Online matching problems have garnered significant attention in recent years due to numerous applications in e-commerce, online advertisements, ride-sharing, etc. Many of them capture the uncertainty in the real world by including stochasticity in both the arrival process and the matching process. The Online Stochastic Matching with Timeouts problem introduced by Bansal, et al., (Algorithmica, 2012) models matching markets (e.g., E-Bay, Amazon). Buyers arrive from an independent and identically distributed (i.i.d.) known distribution on buyer profiles and can be shown a list of items one at a time. Each buyer has some probability of purchasing each item and a limit (timeout) on the number of items they can be shown. Bansal et al., (Algorithmica, 2012) gave a 0.12-competitive algorithm which was improved by Adamczyk, et al., (ESA, 2015) to 0.24. We present an online attenuation framework that uses an algorithm for offline stochastic matching as a black box. On the upper bound side, we show that this framework, combined with a black-box adapted from Bansal et al., (Algorithmica, 2012), yields an online algorithm which nearly doubles the ratio to 0.46. On the lower bound side, we show that no algorithm can achieve a ratio better than 0.632 using the standard LP for this problem. This framework has a high potential for further improvements since new algorithms for offline stochastic matching can directly improve the ratio for the online problem. Our online framework also has the potential for a variety of extensions. For example, we introduce a natural generalization: Online Stochastic Matching with Two-sided Timeouts in which both online and offline vertices have timeouts. Our framework provides the first algorithm for this problem achieving a ratio of 0.30. We once again use the algorithm of Adamczyk et al., (ESA, 2015) as a black-box and plug-it into our framework.

研究动机与目标

• 解决在线随机匹配与超时（OSMT）问题，其中在线买家具有有限耐心，且物品购买具有概率性。
• 通过引入一种通用的基于衰减的框架，将竞争比提升至超越以往0.24的水平。
• 将该框架扩展至更复杂的双侧超时场景，即在线顶点和离线顶点均具有耐心约束。
• 提供一种统一且可扩展的方法，利用离线随机匹配领域的进展来提升在线性能。
• 为标准线性规划（LP）建立0.632的理论下界，表明在当前公式下改进存在固有极限。

提出的方法

• 该框架采用局部衰减：通过全局衰减参数α缩放探测边的概率，以减少在线到达之间的干扰。
• 提出两种主要在线框架，即ATTN₁和ATTN₂，它们在将离线黑箱解部署到在线环境前应用衰减。
• 该方法依赖于黑箱归约：任何适用于星形图上离线随机匹配的算法均可被集成到该框架中，从而生成在线算法。
• 分析过程使用条件概率和马尔可夫不等式，以界定物品保持未匹配状态的概率，从而在衰减条件下确保高期望收益。
• 该框架被扩展以处理分数到达率，并通过修改探测与承诺规则，推广至双侧超时场景。
• 通过仔细分析衰减机制下边的探测概率与匹配事件，证明了竞争比。

实验结果

研究问题

• RQ1 能否设计一种通用框架，通过利用离线算法来提升在线随机匹配与超时问题的竞争比？
• RQ2 在在线环境中对离线解应用衰减后，可实现多高的竞争比？
• RQ3 该框架能否扩展至处理双侧超时问题，即在线与离线顶点均具有耐心约束？
• RQ4 基于标准线性规划公式，该问题的理论改进极限是什么？
• RQ5 该框架如何通过更优的离线黑箱算法实现未来改进？

主要发现

• ATTN₁框架结合Bansal等人（2012）提出的黑箱算法，实现了0.46的竞争比，几乎将以往最佳水平0.24提升了一倍。
• 在双侧超时问题中，该框架首次实现了常数竞争比，达到0.30。
• 该框架具有可扩展性：针对星形图的更优离线算法可直接带来更优的在线性能。
• 为标准LP建立了0.632的下界，表明在当前公式下，任何算法的竞争比均无法超过此值。
• 分析证明，在该框架下，物品被匹配的概率下限为αe⁻ᵅ倍的离线比例，当最优α = 0.5时，广义模型下的竞争比可达0.303。