Approximation schemes for generalized two-dimensional vector packing with application to data placement

Given is a set of items and a set of devices, each possessing two limited resources. Each item requires some amounts of these resources. Further, each item is associated with a profit and a color, and items of the same color can share the use of one resource. The goal is to allocate the resources to the most profitable (feasible) subset of items. In alternative formulation, the goal is to pack the most profitable subset of items in a set of two-dimensional bins (knapsacks), in which the capacity in one dimension is sharable. Indeed, the special case where there is a single item in each color is the well-known two-dimensional vector packing (2DVP) problem. Thus, unless P = NP, the problem that we study does not admit a fully polynomial time approximation scheme (FPTAS) for a single bin, and is MAX-SNP hard for multiple bins. Our problem has several important applications, including data placement on disks in media-on-demand systems. We present approximation algorithms as well as optimal solutions for some instances. In some cases, our results are similar to the best known results for 2DVP. Specifically, for a single bin, we show that the problem is solvable in pseudo-polynomial time and develop a polynomial time approximation scheme (PTAS) for general instances. For a natural subclass of instances we obtain a simpler scheme. This yields the first combinatorial PTAS for a non-trivial subclass of instances for 2DVP. For multiple bins, we develop a PTAS for a subclass of instances arising in the data placement problem. Finally, we show that when the number of distinct colors in the instance is fixed, our problem admits a PTAS, even if the items have arbitrary sizes and profits, and the bins are arbitrary.