Online Matching with Delays and Size-based Costs
Yasushi Kawase, Tomohiro Nakayoshi
TL;DR
This work studies Online Matching with Delays and Size-based Costs (OMDSC), where requests arrive over time and can be formed into groups whose total cost combines a size-based penalty and waiting costs. The authors classify binary penalty functions by their zero-penalty sets $B_f$ into three cases, deriving tight results: (i) when $B_f=\,\emptyset$, the problem reduces to the TCP acknowledgment problem with a $2$-competitive algorithm; (ii) when $B_f=\\{kn: n\in\mathbb{Z}_{++}\}$, they show that naive match-all-remaining strategies are $\Omega(\sqrt{k})$-competitive, but there exists a deterministic $O(\log k/\log\log k)$-competitive algorithm, and this bound is optimal; (iii) when $B_f$ is neither empty nor multiples, no online algorithm can achieve a finite competitive ratio. For the specific case (ii), the paper provides both a constructive upper bound via a phased, phase-aware algorithm using a parameter $\alpha$ solving $\alpha^\alpha=k$ (yielding $O(\log k/\log\log k)$) and a matching lower bound, establishing tightness. The results illuminate the trade-offs between group size, waiting costs, and non-monotone penalties in online batching problems with practical implications for batch processing in online services and game matchmaking. Overall, the work advances the theory of online matching with delays by precisely characterizing when competitive algorithms exist and how tight the attainable ratios can be under different penalty structures.
Abstract
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.