Online Bin Packing with Known T

In the online bin packing problem, a sequence of items is revealed one at a time, and each item must be packed into an available bin instantly upon its arrival. In this paper, we revisit the problem under a setting where the total number of items T is known in advance, also known as the closed online bin packing problem. Specifically, we study both the stochastic model where the item sizes are drawn independently from an unknown distribution and the random permutation model where the item sizes may be adversarially chosen, but the items arrive in a randomly permuted order. We develop and analyze an adaptive algorithm that solves an offline bin packing problem at geometric time intervals and uses the offline optimal solution to guide the online packing decisions. Under both models, we show that the algorithm achieves C √ T regret (in terms of the number of used bins) compared to the hindsight optimal solution, where C is a universal constant (≤ 13) that bears no dependence on the underlying distribution or the item sizes. The result shows the lower bound barrier of Ω( √ T log T ) in (Shor, 1986) can be broken with solely the knowledge of the horizon T , but without a need of knowing the underlying distribution. As to the algorithm analysis, we develop a new approach to analyzing the packing dynamic using the notion of exchangeable random variables. The approach creates a symmetrization between the offline solution and the online solution, and it is used to analyze both the algorithm performance and the various benchmarks related to the bin packing problem. For the latter one, our analysis provides an alternative (probably simpler) treatment and tightens the analysis of the asymptotic benchmark of the stochastic bin packing problem in (Rhee and Talagrand, 1989a,b). As the analysis only relies on a symmetry between the offline and online problems, the algorithm and benchmark analyses can be naturally extended from the stochastic model to the random permutation model.