Optimal Analysis of Best Fit Bin Packing

Abstract

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 optimum needs Opt bins, BestFit always uses at most \(\lfloor1.7\cdot\) OPT \(\rfloor\) bins. Furthermore we show matching lower bounds for all values of Opt, i.e., we give instances on which BestFit uses exactly \(\lfloor1.7\cdot\) OPT \(\rfloor\) bins. Thus we completely settle the worst-case complexity of BestFit bin packing after more than 40 years of its study.