Vector bin packing with multiple-choice

We consider a variant of bin packing called multiple-choice vector bin packing  . In this problem, we are given a set of nn items, where each item can be selected in one of several DD-dimensional incarnations  . We are also given TT bin types, each with its own cost andDD-dimensional size. Our goal is to pack the items in a set of bins of minimum overall cost. The problem is motivated by scheduling in networks with guaranteed quality of service (QoS), but due to its general formulation it has many other applications as well. We present an approximation algorithm that is guaranteed to produce a solution whose cost is about lnDlnD times the optimum. For the running time to be polynomial we require D=O(1)D=O(1) and T=O(logn)T=O(logn). This extends previous results for vector bin packing, in which each item has a single incarnation and there is only one bin type. To obtain our result we also present a PTAS for the multiple-choice version of multidimensional knapsack, where we are given only one bin and the goal is to pack a maximum weight set of (incarnations of) items in that bin.