Online TCP Acknowledgment under General Delays

In a seminal work, Dooly, Goldman, and Scott (STOC 1998; JACM 2001) introduced the classic Online TCP Acknowledgment problem. In this problem, a sequence of $n$ packets arrives over time, and the objective is to minimize both the number of acknowledgments sent and the total delay experienced by the packets. They showed that a greedy algorithm -- acknowledge when the delay of pending packets equals the acknowledgment cost -- is $2$-competitive. Online TCP Acknowledgment is the canonical online problem with delay, capturing the fundamental tradeoff between reducing service cost through batching and the delay incurred by pending requests. Prior work has largely focused on different types of service costs, e.g., Joint Replenishment. However, to the best of our knowledge, beyond the work of Albers and Bals (SODA 2003), which studies maximum delay and closely related objectives, not much is known for more general delay cost models. In this work, we study the Online TCP Acknowledgment under two new generalized delay costs that we call batch-oblivious and batch-aware. In the former, the delay cost is a function of the vector of packet delays. We show that the classic greedy approach is also $2$-competitive for continuous submodular functions and $\ell_p$ norms. In the batch-aware model, each batch incurs a delay cost that is a function $f$ of the vector of delays incurred by its packets. When the overall delay cost is the maximum of the batch delay costs, we show that the greedy approach is also $2$-competitive. Our main technical contribution is for the batch-aware setting, where the overall delay cost is the sum of the batch delay costs. We show that the greedy approach is $\Omega(n)$-competitive and that the deterministic competitive ratio is $\Theta(\log n)$. Remarkably, our algorithm only requires the bare minimum assumption that $f$ is monotone.