Online Matching with Size-Based and Convex Delays

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. Our bounds are in terms of $n$, as opposed to Deryckere and Umboh's bounds that depend on $m$. 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 $\Omega(n)$ 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.