[Paper Review] 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 Problems24 references3 citations

TL;DR

This paper introduces an attenuation-based online framework for stochastic matching with timeouts, using an offline stochastic matching algorithm as a black box. It achieves a 0.46-competitive ratio for the standard problem and the first 0.30 ratio for two-sided timeouts, significantly improving prior results while enabling future gains via better offline black boxes.

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.

Motivation & Objective

• To address the Online Stochastic Matching with Timeouts (OSMT) problem, where online buyers have limited patience and item purchases are probabilistic.
• To improve the competitive ratio beyond prior work, which had reached 0.24, by introducing a general attenuation-based framework.
• To extend the framework to the more complex setting of two-sided timeouts, where both online and offline vertices have patience constraints.
• To provide a unified, extensible approach that leverages advances in offline stochastic matching to improve online performance.
• To establish a theoretical lower bound of 0.632 for the standard LP, indicating inherent limits to improvement.

Proposed method

• The framework uses local attenuation: it scales down the probability of probing edges based on a global attenuation parameter α to reduce interference between online arrivals.
• It introduces two main online frameworks, ATTN₁ and ATTN₂, which apply attenuation to the offline black-box solution before deploying it in the online setting.
• The method relies on a black-box reduction: any algorithm for offline stochastic matching on star graphs can be plugged into the framework to yield an online algorithm.
• The analysis uses conditional probability and Markov's inequality to bound the probability that an item remains unmatched, ensuring high expected reward under attenuation.
• The framework is extended to handle fractional arrival rates and generalizes to two-sided timeouts by modifying the probing and commitment rules.
• It proves competitive ratios via careful analysis of edge probing probabilities and matching events under the attenuation mechanism.

Experimental results

Research questions

• RQ1 Can a general framework be designed to improve competitive ratios in online stochastic matching with timeouts by leveraging offline algorithms?
• RQ2 What competitive ratio can be achieved by applying attenuation to the offline solution in the online setting?
• RQ3 Can the framework be extended to handle two-sided timeouts, where both online and offline vertices have patience constraints?
• RQ4 What is the theoretical limit of improvement for this problem, given the standard LP formulation?
• RQ5 How does the framework enable future improvements through better offline black-box algorithms?

Key findings

• The ATTN₁ framework combined with a black-box adapted from Bansal et al. (2012) achieves a 0.46-competitive ratio, nearly doubling the prior best of 0.24.
• For the two-sided timeouts problem, the framework yields the first constant-competitive algorithm with a 0.30 competitive ratio.
• The framework is extensible: improved offline algorithms for star graphs directly lead to better online performance.
• A lower bound of 0.632 is established for the standard LP, indicating that no algorithm can exceed this ratio under the current formulation.
• The analysis proves that the probability of matching an item under the framework is bounded below by αe⁻ᵅ times the offline fraction, with optimal α = 0.5 yielding 0.303 for the generalized model.