We study a natural generalization of the Unsplittable Flow Problem (UFP) on a path. We generalize the problem by allowing multiple parallel paths (machines). The objective is to find a profit-maximizing feasible assignment of n given tasks to m machines. Each task has a given machine-independent start and finish time, as well as a profit and a resource demand. Each machine is characterized by a given time-varying capacity. At any point in time, the total demand of tasks assigned to a machine must not exceed its available capacity. To the best of our knowledge, this is the first work to study the multiple path generalization of the UFP on a path.

Assuming that all machines have uniform capacities and the number of machines is constant, we present a polynomial-time (2 + ϵ)-approximation algorithm for this NP-hard problem, derived by combining randomized rounding techniques with dynamic programming. Furthermore, under the no-bottleneck assumption (NBA) and assuming that the tasks are small, i.e., their demand-to-capacity ratio is less than a fixed value δ between their start and finish times for all the machines, we present a randomized-rounding-based polynomial-time constant factor approximation algorithm.

Additionally, for the case of uniform capacities where demands are at most half of the capacity, we present a generalization of the "List Algorithm" given in [2]. We prove that this LP-rounding-based method achieves a polynomial-time 3-approximation. This result extends the 2-approximation from the single-machine setting, with the increased ratio reflecting the additional disjointness constraints required when assigning tasks across multiple parallel machines. Finally, we exhibit instances with unit demands that possess an integrality gap greater than 1, a phenomenon that does not occur in the standard UFP on a path (i.e., the single machine case).