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 + ε) -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 provide a (10 + ε) -approximation algorithm for SAP on ring networks.