A constant factor approximation algorithm for the storage allocation problem

We study the storage allocation problem (sap) which is a variant of the unsplittable flow problem on paths (ufpp). a sap instance consists of a path P = (V, E) and a set J of tasks. Each edge e ⋯ E has a capacity c _e and each task j ⋯ J is associated with a path I_j in P, a demand d_j and a weight W_j. The goal is to find a maximum weight subset S ⊆ J of tasks and a height function h : S → ℝ⁺ such that (i) h(j)+d_j ≤ c_e, for every e ⋯ I_J; and (ii) if j, i ⋯ S such that I _j ∩ I_i ≠ and h (j) ≥ h(i), then h(J) ≥ h(i) + d_i. SAP can be seen as a rectangle packing problem in which rectangles can be moved vertically, but not horizontally. We present a polynomial time (9 + e)-approximation algorithm for SAP. Our algorithm is based on a variation of the framework for approximating UFPP by Bonsma et al. [FOCS 2011] and on a (4 + ε)-approximation algorithm for δ-small SAP instances (in which d_j ≤ δ · c_e, for every e ⋯ I_j, for a sufficiently small constant δ > 0). In our algorithm for δ-small instances, tasks are packed carefully in strips in a UFPP manner, and then a (1 + ε) factor is incurred by a reduction from SAP to UFPP in strips. The strips are stacked to form a SAP solution. Finally, we show that SAP is strongly NP-hard, even with uniform weights and even if assuming the no bottleneck assumption.