Suppose that we have a set of items and a set of devices, each possessing two limited resources. Each item requires a given amount of the 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. We need to allocate the resources to the most profitable (feasible) subset of items. In alternative formulation, we need to pack the most profitable subset of items in a set of 2-dimensional bins (knapsacks), in which the capacity in one dimension is shamble. Indeed, the special case where we have a single item in each color is the well-known 2-dimensional vector packing (2DVP) problem. Thus, the problem that we study is strongly NP-hard for a single bin, and 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 knapsack, we show that our 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 knapsacks, 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.
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