Optimal Analysis of Best Fit Bin Packing

In the bin packing problem we are given an instance consisting of a sequence of items with sizes between 0 and 1. The objective is to pack these items into the smallest possible number of bins of unit size. BestFit algorithm packs each item into the most full bin where it fits, possibly opening a new bin if the item cannot fit into any currently open bin. In early seventies it was shown that the asymptotic approximation ratio of BestFit bin packing is equal to 1.7. We prove that also the absolute approximation ratio for BestFit bin packing is exactly 1.7, improving the previous bound of 1.75. This means that if the optimumneeds Opt bins, BestFit always uses at most b1.7 ·OPTc bins. Furthermore we show matching lower bounds for all values of Opt, i.e., we giveinstances on which BestFit uses exactly b1.7 ·OPTc bins. Thus we completely settle the worst-case complexity of BestFit bin packing after more than 40 years of its study. ∗Department of Mathematics, University of Pannonia, Veszprém, Hungary, [email protected] by the Hungarian State and the European Union under the TAMOP-4.2.2.A-11/1/ KONV-2012-0072.†Computer Science Institute of Charles University, Faculty of Mathematics and Physics, Malostranskénám. 25, CZ-11800 Praha 1, Czech Republic, [email protected]. Partially supported by theCenter of Excellence – Inst. for Theor. Comp. Sci., Prague (project P202/12/G061 of GA ČR) and grantIAA100190902 of GA AV ČR.