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.

In the Min-cost Perfect Matching with Delays (MPMD) problem, 2m requests arrive over time at points of a metric space. An online algorithm has to connect these requests in pairs, but a decision to match may be postponed till a more suitable matching pair is found. The goal is to minimize the joint cost of connection and the total waiting time of all requests.We present an O(m)-competitive deterministic algorithm for this problem, improving on an existing bound of O(m log 2 5.5 ) = O(m 2.46 ). Our algorithm also solves (with the same competitive ratio) a bipartite variant of MPMD, where requests are either positive or negative and only requests with different polarities may be matched with each other. Unlike the existing randomized solutions, our approach does not depend on the size of the metric space and does not have to know it in advance. ACM Subject Classification Theory of computation → Online algorithms

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