In this paper, we introduce the problem of Online Matching with Delays and Size-based Costs (OMDSC). The OMDSC problem involves $m$ requests arriving online. At any time, a group can be formed by matching any number of these requests that have been received but are still unmatched. The cost associated with each group is determined by the waiting time for each request within the group and a size-dependent cost. Our goal is to partition all incoming requests into multiple groups while minimizing the total associated cost. The problem extends the TCP acknowledgment problem proposed by Dooly et al. (JACM 2001). It generalizes the cost model for sending acknowledgments. This paper reveals the competitive ratios for a fundamental case where the range of the penalty function is limited to $0$ and $1$. We classify such penalty functions into three distinct cases: (i) a fixed penalty of $1$ regardless of group size, (ii) a penalty of $0$ if and only if the group size is a multiple of a specific integer $k$, and (iii) other situations. The problem of case (i) is equivalent to the TCP acknowledgment problem, for which Dooly et al. proposed a $2$-competitive algorithm. For case (ii), we first show that natural algorithms that match all the remaining requests are $\Omega(\sqrt{k})$-competitive. We then propose an $O(\log k / \log \log k)$-competitive deterministic algorithm by carefully managing match size and timing, and we also prove its optimality. For case (iii), we demonstrate the non-existence of a competitive online algorithm. Additionally, we discuss competitive ratios for other typical penalty functions.

翻译：本文提出了具有延迟与规模相关成本的在线匹配问题。该问题涉及$m$个在线到达的请求。在任意时刻，可将已到达但尚未匹配的任意数量请求进行分组匹配。每个分组的成本由组内各请求的等待时间及与分组规模相关的惩罚成本共同决定。我们的目标是将所有到达请求划分至多个分组，同时最小化总成本。该问题扩展了Dooly等人提出的TCP确认问题，并推广了发送确认的成本模型。本文针对惩罚函数值域限定为$0$和$1$的基础情形揭示了竞争比。我们将此类惩罚函数分为三种情况：(i) 无论分组规模大小，惩罚值恒为$1$；(ii) 当且仅当分组规模为特定整数$k$的倍数时惩罚值为$0$；(iii) 其他情形。情况(i)等价于TCP确认问题，Dooly等人已提出$2$-竞争算法。对于情况(ii)，我们首先证明直接匹配所有剩余请求的自然算法具有$\Omega(\sqrt{k})$竞争比，随后通过精细管理匹配规模与时机，提出$O(\log k / \log \log k)$-竞争确定性算法并证明其最优性。对于情况(iii)，我们证明了竞争性在线算法不存在性。此外，我们还讨论了其他典型惩罚函数的竞争比。