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

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 $2 m$ 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

Min-Cost Bipartite Perfect Matching with Delays

In the min-cost bipartite perfect matching with delays (MBPMD) problem, requests arrive online at points of a finite metric space. Each request is either positive or negative and has to be matched to a request of opposite polarity. As opposed to traditional online matching problems, the algorithm does not have to serve requests as they arrive, and may choose to match them later at a cost. Our objective is to minimize the sum of the distances between matched pairs of requests (the connection cost) and the sum of the waiting times of the requests (the delay cost). This objective exhibits a natural tradeoff between minimizing the distances and the cost of waiting for better matches. This tradeoff appears in many real-life scenarios, notably, ride-sharing platforms. MBPMD is related to its non-bipartite variant, min-cost perfect matching with delays (MPMD), in which each request can be matched to any other request. MPMD was introduced by Emek et al. (STOC\u2716), who showed an O(log^2(n)+log(Delta))-competitive randomized algorithm on n-point metric spaces with aspect ratio Delta. Our contribution is threefold. First, we present a new lower bound construction for MPMD and MBPMD. We get a lower bound of Omega(sqrt(log(n)/log(log(n)))) on the competitive ratio of any randomized algorithm for MBPMD. For MPMD, we improve the lower bound from Omega(sqrt(log(n))) (shown by Azar et al., SODA\u2717) to Omega(log(n)/log(log(n))), thus, almost matching their upper bound of O(log(n)). Second, we adapt the algorithm of Emek et al. to the bipartite case, and provide a simplified analysis that improves the competitive ratio to O(log(n)). The key ingredient of the algorithm is an O(h)-competitive randomized algorithm for MBPMD on weighted trees of height h. Third, we provide an O(h)-competitive deterministic algorithm for MBPMD on weighted trees of height h. This algorithm is obtained by adapting the algorithm for MPMD by Azar et al. to the apparently more complicated bipartite setting

Set Cover with Delay - Clairvoyance Is Not Required

In most online problems with delay, clairvoyance (i.e. knowing the future delay of a request upon its arrival) is required for polylogarithmic competitiveness. In this paper, we show that this is not the case for set cover with delay (SCD) - specifically, we present the first non-clairvoyant algorithm, which is O(log n log m)-competitive, where n is the number of elements and m is the number of sets. This matches the best known result for the classic online set cover (a special case of non-clairvoyant SCD). Moreover, clairvoyance does not allow for significant improvement - we present lower bounds of ?(?{log n}) and ?(?{log m}) for SCD which apply for the clairvoyant case. In addition, the competitiveness of our algorithm does not depend on the number of requests. Such a guarantee on the size of the universe alone was not previously known even for the clairvoyant case - the only previously-known algorithm (due to Carrasco et al.) is clairvoyant, with competitiveness that grows with the number of requests. For the special case of vertex cover with delay, we show a simpler, deterministic algorithm which is 3-competitive (and also non-clairvoyant)

Deterministic Primal-Dual Algorithms for Online k-way Matching with Delays

In this paper, we study the Min-cost Perfect $k$-way Matching with Delays ($k$-MPMD), recently introduced by Melnyk et al. In the problem, $m$ requests arrive one-by-one over time in a metric space. At any time, we can irrevocably make a group of $k$ requests who arrived so far, that incurs the distance cost among the $k$ requests in addition to the sum of the waiting cost for the $k$ requests. The goal is to partition all the requests into groups of $k$ requests, minimizing the total cost. The problem is a generalization of the min-cost perfect matching with delays (corresponding to $2$-MPMD). It is known that no online algorithm for $k$-MPMD can achieve a bounded competitive ratio in general, where the competitive ratio is the worst-case ratio between its performance and the offline optimal value. On the other hand, $k$-MPMD is known to admit a randomized online algorithm with competitive ratio $O(k^{5}\log n)$ for a certain class of $k$-point metrics called the $H$-metric, where $n$ is the size of the metric space. In this paper, we propose a deterministic online algorithm with a competitive ratio of $O(mk^2)$ for the $k$-MPMD in $H$-metric space. Furthermore, we show that the competitive ratio can be improved to $O(m + k^2)$ if the metric is given as a diameter on a line

Online Matching with Set Delay

We initiate the study of online problems with set delay, where the delay cost at any given time is an arbitrary function of the set of pending requests. In particular, we study the online min-cost perfect matching with set delay (MPMD-Set) problem, which generalises the online min-cost perfect matching with delay (MPMD) problem introduced by Emek et al. (STOC 2016). In MPMD, $m$ requests arrive over time in a metric space of $n$ points. When a request arrives the algorithm must choose to either match or delay the request. The goal is to create a perfect matching of all requests while minimising the sum of distances between matched requests, and the total delay costs incurred by each of the requests. In contrast to previous work we study MPMD-Set in the non-clairvoyant setting, where the algorithm does not know the future delay costs. We first show no algorithm is competitive in $n$ or $m$. We then study the natural special case of size-based delay where the delay is a non-decreasing function of the number of unmatched requests. Our main result is the first non-clairvoyant algorithms for online min-cost perfect matching with size-based delay that are competitive in terms of $m$. In fact, these are the first non-clairvoyant algorithms for any variant of MPMD. Furthermore, we prove a lower bound of $\Omega(n)$ for any deterministic algorithm and $\Omega(\log n)$ for any randomised algorithm. These lower bounds also hold for clairvoyant algorithms

Online Matching with Set and Concave Delays

We initiate the study of online problems with set delay, where the delay cost at any given time is an arbitrary function of the set of pending requests. In particular, we study the online min-cost perfect matching with set delay (MPMD-Set) problem, which generalises the online min-cost perfect matching with delay (MPMD) problem introduced by Emek et al. (STOC 2016). In MPMD, m requests arrive over time in a metric space of n points. When a request arrives the algorithm must choose to either match or delay the request. The goal is to create a perfect matching of all requests while minimising the sum of distances between matched requests, and the total delay costs incurred by each of the requests. In contrast to previous work we study MPMD-Set in the non-clairvoyant setting, where the algorithm does not know the future delay costs. We first show no algorithm is competitive in n or m. We then study the natural special case of size-based delay where the delay is a non-decreasing function of the number of unmatched requests. Our main result is the first non-clairvoyant algorithms for online min-cost perfect matching with size-based delay that are competitive in terms of m. In fact, these are the first non-clairvoyant algorithms for any variant of MPMD. A key technical ingredient is an analog of the symmetric difference of matchings that may be useful for other special classes of set delay. Furthermore, we prove a lower bound of ?(n) for any deterministic algorithm and ?(log n) for any randomised algorithm. These lower bounds also hold for clairvoyant algorithms. Finally, we also give an m-competitive deterministic algorithm for uniform concave delays in the clairvoyant setting

The Min-Cost Matching with Concave Delays Problem

We consider the problem of online min-cost perfect matching with concave delays. We begin with the single location variant. Specifically, requests arrive in an online fashion at a single location. The algorithm must then choose between matching a pair of requests or delaying them to be matched later on. The cost is defined by a concave function on the delay. Given linear or even convex delay functions, matching any two available requests is trivially optimal. However, this does not extend to concave delays. We solve this by providing an $O(1)$-competitive algorithm that is defined through a series of delay counters. Thereafter we consider the problem given an underlying $n$-points metric. The cost of a matching is then defined as the connection cost (as defined by the metric) plus the delay cost. Given linear delays, this problem was introduced by Emek et al. and dubbed the Min-cost perfect matching with linear delays (MPMD) problem. Liu et al. considered convex delays and subsequently asked whether there exists a solution with small competitive ratio given concave delays. We show this to be true by extending our single location algorithm and proving $O(\log n)$ competitiveness. Finally, we turn our focus to the bichromatic case, wherein requests have polarities and only opposite polarities may be matched. We show how to alter our former algorithms to again achieve $O(1)$ and $O(\log n)$ competitiveness for the single location and for the metric case.Comment: 40 pages, 4 figure

Online Metric Matching with Delay

Traditionally, an online algorithm must service a request upon its arrival. In many practical situations, one can delay the service of a request in the hope of servicing it more efficiently in the near future. As a result, the study of online algorithms with delay has recently gained considerable traction. For most online problems with delay, competitive algorithms have been developed that are independent of properties of the delay functions associated with each request. Interestingly, this is not the case for the online min-cost perfect matching with delays (MPMD) problem, introduced by Emek et al.(STOC 2016). In this thesis we show that some techniques can be modified to extend to larger classes of delay functions, without affecting the competitive ratio. In the interest of designing competitive solutions for the problem in a more general setting, we introduce the study of online problems with set delay. Here, the delay cost at any time is given by an arbitrary function of the set of pending requests, rather than the sum over individual delay functions associated with each request. In particular, we study the online min-cost perfect matching with set delay (MPMD-Set) problem, which provides a generalisation of MPMD. In contrast to previous work, the new model allows us to study the problem in the non-clairvoyant setting, i.e. where the future delay costs are unknown to the algorithm. We prove that for MPMD-Set in the most general non-clairvoyant setting, there exists no competitive algorithm. Motivated by this impossibility, we introduce a new class of delay functions called sizebased and prove that for this version of the problem, there exist both non-clairvoyant deterministic and randomised algorithms that are competitive in the number of requests. Our results reveal that the quality of an online matching depends both on the algorithm's access to information about future delay costs, and the properties of the delay function