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 delta between the demand of i and the minimum capacity of any edge used by i. If all tasks have sufficiently small delta, the problem is already well understood and there is a 1+eps approximation. For the complementary setting---instances whose tasks all have large delta---much remains unknown, and the best known polynomial-time procedure gives only (for any constant delta>0) an approximation ratio of 6+eps.   In this paper we present a polynomial time 1+eps 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 delta-small tasks, our result implies a 2+eps approximation for UFP, improving on the previous best 7+eps 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. 
Aris Anagnostopoulos, Fabrizio Grandoni, Stefano Leonardi, and Andreas   Wiese
cs.DS