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.