Attenuate Locally, Win Globally: Attenuation-Based Frameworks for Online Stochastic Matching with Timeouts

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 and matching processes. 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 several online attenuation frameworks that use an algorithm for offline stochastic matching as a black box. On the upper bound side, we show that one 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. Additionally, our attenuation frameworks extend to the more general setting of fractional arrival rates for online vertices. 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 frameworks also have 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 frameworks provide the first algorithm for this problem achieving a ratio of 0.30. We once again use the algorithm of Bansal et al., (Algorithmica, 2012) as a black-box and plug it into one of our frameworks. * This is the full version of the paper that appeared in AAMAS-2017 [10]. There was an error in one of the offline black-box. This version fixes all the ratios in the theorems.

Multi-Stage and Multi-Customer Assortment Optimization With Inventory Constraints

“…First, in the special case of online stochastic matching with timeouts, our algorithm imitates a scaled copy of the LP solution using a randomized rounding scheme. We improve previous schemes by increasing the likelihood that items with lower sales probabilities are offered first, and then use the framework of Brubach et al (2017) to bound the probability that each item has positive remaining inventory.…”

“…As discussed earlier in this section, attenuation framework is used to address the online subproblem. Brubach et al (2017) provide three different attenuation frameworks: the edge-attenuation, the vertex-attenuation and the combined edge and vertex-attenuation. The last one is the most powerful one, giving Brubach et al (2017) the approximation ratio of 0.46 for the online stochastic matching with timeouts problem.…”

“…Brubach et al (2017) provide three different attenuation frameworks: the edge-attenuation, the vertex-attenuation and the combined edge and vertex-attenuation. The last one is the most powerful one, giving Brubach et al (2017) the approximation ratio of 0.46 for the online stochastic matching with timeouts problem. Thus, we only study the performance of our online algorithm when the combined edge and vertex-attenuation framework (called the attenuation framework henceforth) is used alongside the offline black-box.…”

“…In this case, the firm shows the items to the customer one-at-a-time, which gives the firm increased control and is also natural in some applications such as online dating. This special case corresponds to the online stochastic matching with timeouts problem, which has been studied in a stream of work (Bansal et al 2012, Adamczyk et al 2015, Brubach et al 2017) originating from Chen et al (2009).…”

“…At this point, we invoke (Brubach et al 2017, Lemma 3.1) from their original attenuation framework, which implies that at time-step t, item i being available and item i ′ being available are negatively correlated. Therefore, we can subsequently derive…”