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’16), who showed an O(log2 n + log ∆)-competitive randomized algorithm on n-point metric spaces with aspect ratio ∆. Our contribution is threefold. First, we present a new lower bound construction for MPMD and MBPMD. We get a lower bound of Ω (√ log n log log n ) on the competitive ratio of any randomized algorithm for MBPMD. For MPMD, we improve the lower bound from Ω( √ logn) (shown by Azar et al., SODA’17) to Ω ( log n log log n ) , thus, almost matching their upper bound of O(logn). Second, we adapt the algorithm of Emek et al. to the bipartite case, and provide a simplified analysis ∗ Yossi Azar and Ashish Chiplunkar were supported in part by the Israel Science Foundation (grant no. 1506/16), by the I-CORE program (center no. 4/11), and by the Blavatnik Fund. © Itai Ashlagi, Yossi Azar, Moses Charikar, Ashish Chiplunkar, Ofir Geri, Haim Kaplan, Rahul Makhijani, Yuyi Wang, and Roger Wattenhofer; licensed under Creative Commons License CC-BY Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Editors: Klaus Jansen, José D.P. Rolim, David Williamson, and Santosh S. Vempala; Article No. 1; pp. 1:1–1:20 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 1:2 Min-Cost Bipartite Perfect Matching with Delays that improves the competitive ratio to O(logn). 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. 1998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity, F.1.2 [Modes of Computation] Online Computation