Packing Resizable Items with Application to Video Delivery over Wireless Networks

Abstract

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 > 0, and a set I of items. Each item j ∈ I is of size s j > 0. A packed item must stay in the bin for a fixed time interval. To accommodate more items in the bin, each item j can be compressed to a size p j ∈ [0, s j) for at most a fraction q j ∈ [0, 1) of the packing interval. The goal is to pack in the bin, for the given time interval, a subset of items of maximum cardinality. PRI 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 2 3 OP T (I) − 3 items, where OP T (I) is the number of items packed in an optimal solution. Our algorithm yields better ratio for instances in which the maximum compression time of an item is q max ∈ (0, 1 2). For subclasses of instances arising in realistic scenarios, we give an algorithm that packs at least OP T (I) − 2 items. Finally, we show that a non-trivial subclass of instances admits an asymptotic fully polynomial time approximation scheme (AFPTAS).