On the Approximability of Unsplittable Flow on a Path with Time Windows

Alexander Armbruster; Fabrizio Grandoni; Edin Husić; Antoine Tinguely; Andreas Wiese

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arXiv · 2503.17802

On the Approximability of Unsplittable Flow on a Path with Time Windows

Alexander Armbruster·Fabrizio Grandoni·Edin Husić·Antoine Tinguely·Andreas Wiese

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Abstract

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.

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Alexander Armbruster, Fabrizio Grandoni, Edin Husić, Antoine Tinguely, Andreas Wiese. 2025-03-22. On the Approximability of Unsplittable Flow on a Path with Time Windows. https://arxiv.org/abs/2503.17802

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