Beating the Harmonic Lower Bound for Online Bin Packing

In the online bin packing problem, items of sizes in (0,1] arrive online to be packed into bins of size 1. The goal is to minimize the number of used bins. Harmonic++ achieves a competitive ratio of 1.58889 and belongs to the Super Harmonic framework [Seiden, J. ACM, 2002]; a lower bound of Ramanan et al. shows that within this framework, no competitive ratio below 1.58333 can be achieved [Ramanan et al., J. Algorithms, 1989]. In this paper, we present an online bin packing algorithm with asymptotic performance ratio of 1.5815, which constitutes the first improvement in fifteen years and reduces the gap to the lower bound by roughly 15%. We make two crucial changes to the Super Harmonic framework. First, some of the decisions of the algorithm will depend on exact sizes of items, instead of only their types. In particular, for item pairs where the size of one item is in (1/3,1/2] and the other is larger than 1/2 (a large item), when deciding whether to pack such a pair together in one bin, our algorithm does not consider their types, but only checks whether their total size is at most 1. Second, for items with sizes in (1/3,1/2] (medium items), we try to pack the larger items of every type in pairs, while combining the smallest items with large items whenever possible. To do this, we postpone the coloring of medium items (i.e., the decision which items to pack in pairs and which to pack alone) where possible, and later select the smallest ones to be reserved for combining with large items. Additionally, in case such large items arrive early, we pack medium items with them whenever possible. This is a highly unusual idea in the context of Harmonic-like algorithms, which initially seems to preclude analysis (the ratio of items combined with large items is no longer a fixed constant). For the analysis, we carefully mark medium items depending on how they end up packed, enabling us to add crucial constraints to the linear program used by Seiden. We consider the dual, eliminate all but one variable and then solve it with the ellipsoid method using a separation oracle. Our implementation uses additional algorithmic ideas to determine previously hand set parameters automatically and gives certificates for easy verification of the results. We give a lower bound of 1.5766 for algorithms like ours. This shows that fundamentally different ideas will be required to make further improvements