A Match in Time Saves Nine: Deterministic Online Matching with Delays

Abstract

We consider the problem of online Min-cost Perfect Matching with Delays (MPMD) introduced by Emek et al. (STOC 2016). In this problem, an even number of requests appear in a metric space at different times and the goal of an online algorithm is to match them in pairs. In contrast to traditional online matching problems, in MPMD all requests appear online and an algorithm can match any pair of requests, but such decision may be delayed (e.g., to find a better match). The cost is the sum of matching distances and the introduced delays.We present the first deterministic online algorithm for this problem. Its competitive ratio is O(m log 2 5.5 ) = O(m 2.46 ), where 2m is the number of requests. This is polynomial in the number of metric space points if all requests are given at different points. In particular, the bound does not depend on other parameters of the metric, such as its aspect ratio. Unlike previous (randomized) solutions for the MPMD problem, our algorithm does not need to know the metric space in advance. time τ , an online algorithm may decide to match any pair of requests (p i , t i ) and (p j , t j ) that have already arrived (τ ≥ t i and τ ≥ t j ) and have not been matched yet. The cost incurred by such matching edge is dist(p i , p j ) + (τ − t i ) + (τ − t j ), i.e., is the sum of the connection cost and the waiting costs of these two requests.The goal is to eventually match all requests and minimize the total cost. We use a typical yardstick to measure the performance: a competitive ratio [13], defined as the maximum, over all inputs, of the ratios between the cost of an online algorithm and the cost of an optimal offline solution Opt that knows the entire input sequence in advance.

Deterministic Min-Cost Matching with Delays

“…First we consider related work with delays. Since Emek et al [10] introduced the notion of online problems with delayed service, there has been a growing number of works studying such problems (e. [7], are the most closely related to this work. As mentioned above, Emek et al [10] provided a randomized O log 2 n + log ∆ -competitive algorithm for MPMD on general metrics, in which n is the size of the metric space and ∆ is the aspect ratio.…”

“…In online algorithms where one cannot repeat the algorithm in case the cost is high, a deterministic algorithm is preferable. Bienkowski et al [7] provided the first deterministic algorithm for MPMD on general metrics, with a competitive-ratio of O m 2.46 , m being the number of requests. While the previous algorithms require the metric space to be known a priori, their algorithm does not, and is also applicable when the metric space is revealed in an online manner.…”

“…In order to provide a deterministic algorithm, Bienkowski et al [7] used a different approach for the problem -they used a semi-greedy scheme of a ball-growing algorithm. In their analysis, they fix an optimal matching, and charge the cost of each matching-edge generated by their algorithm against the cost of an existing matching-edge of the optimal matching.…”

General Framework for Metric Optimization Problems with Delay or with Deadlines

“…Another metric optimization problem with delay is the problem of matching with delay [2,19,18,4,9,10]. For this problem, arbitrary delay functions are intractable, and thus the main line of work focuses on linear delay functions.…”

A Primal-Dual Online Deterministic Algorithm for Matching with Delays

“…No better lower bounds than the ones used for randomized settings are known for deterministic algorithms. The first solution that worked for general metric spaces was given by Bienkowski et al and achieved an embarrassingly high competitive ratio of O(m log 2 5.5 ) = O(m 2.46 ) [12]. Roughly speaking, their algorithm is based on growing spheres around not-yet-paired requests.…”