A constant-factor approximation algorithm for unsplittable flow on paths

In the unsplittable flow problem on a path, we are given a capacitated path P and n tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks such that, for each edge e of P, the total demand of selected tasks that use e does not exceed the capacity of e. This is a well-studied problem that has been described under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack, and interval packing. We present a polynomial time constant-factor approximation algorithm for this problem. This improves on the previous best known approximation ratio of O(log n). The approximation ratio of our algorithm is 7+ε for any ε < 0. We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves to optimality a special case of the problem of finding a maximum weight independent set of rectangles. In the setting of resource augmentation, wherein the capacities can be slightly violated, we give a (2 + ε)-approximation algorithm. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either 1, 2, or 3.