Improved Approximation Algorithms for Unsplittable Flow on a Path with Time Windows

In the well-studied Unsplittable Flow on a Path problem (UFP), we are given a path graph with edge capacities. Furthermore, we are given a collection of n tasks, each one characterized by a subpath, a weight, and a demand. Our goal is to select a maximum weight subset of tasks so that the total demand of selected tasks using each edge is upper bounded by the corresponding capacity. Chakaravarthy et al. [ESA’14] studied a generalization of UFP, bagUFP, where tasks are partitioned into bags, and we can select at most one task per bag. Intuitively, bags model jobs that can be executed at different times (with different duration, weight, and demand). They gave a \(O(\log n)\) approximation for bagUFP. This is also the best known ratio in the case of uniform weights. In this paper we achieve the following main results: