A Mazing 2+eps Approximation for Unsplittable Flow on a Path

We study the unsplittable flow on a path problem (UFP) where we are given a path with non-negative edge capacities and tasks, which are characterized by a subpath, a demand, and a profit. The goal is to find the most profitable subset of tasks whose total demand does not violate the edge capacities. This problem naturally arises in many settings such as bandwidth allocation, resource constrained scheduling, and interval packing. A natural task classification defines the size of a task i to be the ratio δ between the demand of i and the minimum capacity of any edge used by i. If all tasks have sufficiently small δ, the problem is already well understood and there is a 1 + ε approximation. For the complementary setting— instances whose tasks all have large δ—much remains unknown, and the best known polynomial-time procedure gives only (for any constant δ > 0) an approximation ratio of 6 + ε. In this paper we present a polynomial time 1 + ε approximation for the latter setting. Key to this result is a complex geometrically inspired dynamic program. Here each task is represented as a segment underneath the capacity curve, and we identify a proper maze-like structure so that each passage of the maze is crossed by only O(1) tasks in the computed solution. In combination with the known PTAS for δ-small tasks, our result implies a 2 + ε approximation for UFP, improving on the previous best 7+ε approximation [Bonsma et al., FOCS 2011]. We remark that our improved approximation factor matches the best known approximation ratio for the considerably easier special case of uniform edge capacities.