Tight Bounds for Clairvoyant Dynamic Bin Packing

Abstract

In this paper we focus on the Clairvoyant Dynamic Bin Packing (DBP) problem, which extends the classical online bin packing problem in that items arrive and depart over time and the departure time of an item is known upon its arrival. The problem naturally arises when handling cloud-based networks. We focus specifically on the MinUsageTime cost function which aims to minimize the overall usage time of all bins that are opened during the packing process. Earlier work has shown a O(\frac{\log \mu}{\log \log \mu}) upper bound where \mu is defined as the ratio between the maximal and minimal durations of all items. We improve the upper bound by giving an O(\sqrt{\log \mu})-competitive algorithm. We then provide a matching lower bound of \Omega(\sqrt{\log \mu}) on the competitive ratio of any online algorithm, thus closing the gap with regards to this problem. We then focus on what we call the class of aligned inputs and give a O(\log \log \mu)-competitive algorithm for this case, beating the lower bound of the general case by an exponential factor. Surprisingly enough, the analysis of our algorithm that we present, is closely related to various properties of binary strings.