In the Time-Windows Unsplittable Flow on a Path problem (twUFP) we are given a resource whose available amount changes over a given time interval (modeled as the edge-capacities of a given path $G$) and a collection of tasks. Each task is characterized by a demand (of the considered resource), a profit, an integral processing time, and a time window. Our goal is to compute a maximum profit subset of tasks and schedule them non-preemptively within their respective time windows, such that the total demand of the tasks using each edge $e$ is at most the capacity of $e$. We prove that twUFP is $\mathsf{APX}$-hard which contrasts the setting of the problem without time windows, i.e., Unsplittable Flow on a Path (UFP), for which a PTAS was recently discovered [Grandoni, M\"omke, Wiese, STOC 2022]. Then, we present a quasi-polynomial-time $2+\varepsilon$ approximation for twUFP under resource augmentation. Our approximation ratio improves to $1+\varepsilon$ if all tasks' time windows are identical. Our $\mathsf{APX}$-hardness holds also for this special case and, hence, rules out such a PTAS (and even a QPTAS, unless $\mathsf{NP}\subseteq\mathrm{DTIME}(n^{\mathrm{poly}(\log n)})$) without resource augmentation.

翻译：在带时间窗口的路径上不可分流问题（twUFP）中，我们给定一个资源，其可用量在给定时间区间内变化（建模为给定路径 $G$ 的边容量），以及一组任务。每个任务由需求（所考虑资源的）、收益、整数处理时间和时间窗口来刻画。我们的目标是计算一个最大收益的任务子集，并在各自的时间窗口内非抢占式地调度它们，使得使用每条边 $e$ 的任务总需求不超过 $e$ 的容量。我们证明了 twUFP 是 $\mathsf{APX}$-难的，这与无时间窗口的问题设置（即路径上不可分流问题（UFP））形成对比，后者最近被发现存在 PTAS [Grandoni, M\"omke, Wiese, STOC 2022]。然后，我们在资源增强条件下为 twUFP 提出了一个拟多项式时间的 $2+\varepsilon$ 近似算法。如果所有任务的时间窗口相同，我们的近似比可改进至 $1+\varepsilon$。我们的 $\mathsf{APX}$-难性结果对这一特殊情况也成立，因此排除了在没有资源增强的情况下存在此类 PTAS（甚至 QPTAS，除非 $\mathsf{NP}\subseteq\mathrm{DTIME}(n^{\mathrm{poly}(\log n)})$）的可能性。