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.
翻译:我们以设定的延迟时间开始研究在线问题, 任何特定时间的延迟成本都是一组待决请求的任意功能。 特别是, 我们研究在线的低成本完美与设定的延迟( MPMD- Set) 问题匹配的最小成本完美, 将在线的低成本完美与 Emek 等人( STOC 2016 ) 提出的延迟问题( MPMD ) 相匹配( STOC 2016 ) 。 在 MPMD 中, $ 的请求会以美元为单位, 以美元为单位时不时运到。 当请求到达时, 算法必须选择匹配或延迟请求。 目标是在最小匹配请求和每项请求之间的距离和总延迟费用之间创建一个完美的匹配。 与先前在非cloavovoyan 设置的 MIM- Set 问题相比, 算法没有以美元为单位或美元为单位。 在任何基于规模的延迟情况下, 我们研究一个自然特殊的延迟情况, 在不匹配不匹配的低额的美元请求中, 我们的主要结果是非约束的固定的 。