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. Finally, we also give an $m$-competititve deterministic algorithm for uniform concave delays in the clairvoyant setting.

翻译：我们首次研究了具有集合延迟的在线问题，其中任意时刻的延迟成本是未决请求集合的任意函数。具体而言，我们研究带集合延迟的在线最小代价完美匹配问题，该问题推广了Emek等人提出带延迟的在线最小代价完美匹配问题。在经典问题中，$m$个请求在度量空间的$n$个点上随时间到达。当请求到达时，算法必须选择匹配或延迟该请求。目标是在最小化匹配请求间距离之和以及每个请求产生的总延迟成本的同时，构建所有请求的完美匹配。与先前工作不同，我们在非先知设定下研究该问题，即算法无法预知未来延迟成本。首先，我们证明没有任何算法在$n$或$m$维度上具有竞争性。随后我们研究基于规模的延迟这一自然特例，其中延迟是未匹配请求数量的非递减函数。我们的主要成果是第一个针对基于规模延迟的在线最小代价完美匹配问题的非先知算法，该算法在$m$维度上具有竞争性。事实上，这些也是首个针对任何变体的非先知算法。此外，我们证明了任意确定性算法存在$\Omega(n)$下界，任意随机算法存在$\Omega(\log n)$下界。这些下界同样适用于先知算法。最后，我们在先知设定下为均匀凹延迟给出一个$m$-竞争性确定性算法。