2026-09-10T09:44:54Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:27742 2026-09-09T12:19:36Z ddc:004 open_access

Online Matching with Size-Based and Convex Delays Gan, Junhao Sun, Xiao Umboh, Seeun William online algorithms with delays competitive analysis matching We study the online min-cost perfect matching with delay (MPMD) problem where m requests arrive in a metric space of n points. In MPMD, an algorithm can choose to match a request or to delay, and the objective is to minimise the sum of connection and delay costs. The connection cost of a match is the distance between the locations of two matched requests in the metric, and the increase of the delay cost is a function of the set of unmatched requests at every moment. In this paper, we study two different types of delay functions, size-based (MPMD-Size) and convex delays (MPMD-Convex). The study of MPMD-Size was initiated by Deryckere and Umboh (APPROX/RANDOM 2023) where the instantaneous delay increment is a non-negative monotone function of the number of unmatched requests. We give an exponential improvement on the lower bounds for the deterministic and randomized competitive ratios. We also give improved upper bounds in terms of n, as opposed to Deryckere and Umboh’s upper bounds that were functions of m, which can be much larger than n. Our results settle the deterministic competitive ratio (up to constants). At the heart of these results is a succinct encoding scheme of MPMD-Size on a given n-point metric as a metrical task system problem on a 2^{n-1}-point metric. We also consider MPMD-Convex proposed by Liu et al. (ISAAC 2018) where the delay cost incurred by each request is a uniform convex delay function of the time difference between its arrival time and the moment that it is matched by the algorithm. They focused on delay functions f that are unbounded, non-decreasing, continuous, and satisfy f(0) = f'(0) = 0, and showed that the deterministic competitive ratio is Ω(n) for n-point uniform metrics. We show that, surprisingly, when f is a non-negative, monotone polynomial with f'(0) > 0, there is an O(1)-competitive deterministic algorithm for uniform metrics. Our result completes our understanding of MPMD-Convex on uniform metrics for a broad class of functions. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Junhao Gan and Xiao Sun and Seeun William Umboh 2026 Is Part Of LIPIcs, Volume 392, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2026.25 urn:nbn:de:0030-drops-277428 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.25 eng https://creativecommons.org/licenses/by/4.0/legalcode