Online Matching with Delays and Size-based Costs

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 $Ω(\sqrt{k})$-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.