Packing resizable items with application to video delivery over wireless networks

Motivated by fundamental optimization problems in video delivery over wireless networks, we consider the following problem of packing resizable items (PRI  ). Given is a bin of capacity B>0B>0, and a set I   of items. Each item j∈Ij∈I has a size sj>0sj>0. A packed item must stay in the bin for a fixed common time interval. To accommodate additional items in the bin, each item j can be compressed   to a given size pj∈[0,sj)pj∈[0,sj) for at most a fraction qj∈[0,1)qj∈[0,1) of the packing interval, during one continuous sub-interval. The goal is to pack in the bin, for the given time interval, a subset of items of maximum cardinality. PRI is a generalization of the well-known Knapsack problem, and it is strongly NP-hard already for highly restricted instances.Our main result is an approximation algorithm that packs, for any instance I of PRI  , at least 23OPT(I)−2 items, where OPT(I)OPT(I) is the number of items packed in an optimal solution. Our algorithm yields a better performance ratio for instances in which the maximum compression time of an item is qmax∈(0,12). For subclasses of instances arising in realistic scenarios, we give an algorithm that packs at least OPT(I)−2OPT(I)−2 items. Finally, we show that a non-trivial subclass of instances admits an asymptotic fully polynomial time approximation scheme (AFPTAS).