Online optimization with delays has emerged as a powerful paradigm with numerous applica- tions, including matching, caching, server management, network design, and set cover. In most of these problems, competitive algorithms are known for arbitrary monotone delay functions, even when different requests are associated with different delay functions. The notable exception is online matching with delays, where all previously known algorithms achieving constant or polylogarithmic competitive ratios require the delay functions to be identical, either linear or concave, and known in advance. We study the online minimum-cost perfect matching with delays problem on general metrics. We introduce the broad class of noncrossing delay functions, which strictly generalizes the classic setting of identical concave delay functions while naturally modeling scenarios in which different requests may experience different delay costs. We present the first deterministic $O(1)$-competitive algorithm for the single-location online matching problem with noncrossing delay functions in the non-clairvoyant setting as well as a randomized $O(lognlogΔ+log^2Δ)$-competitive algorithm of an arbitrary metric space where $n$ is the number of points of the metric space and $Δ$ is its aspect ratio. Our result allows each request to have an arbitrary noncrossing delay function, thereby simultaneously removing the assumptions of identical delay functions, concavity, and clairvoyance that were required by all previous constant or polylogarithmic-competitive algorithms.

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