#### An All-Around Near-Optimal Solution for the Classic Bin Packing Problem

##### Shahin Kamali, Alejandro López-Ortiz

In this paper we present the first algorithm with optimal average-case and close-to-best known worst-case performance for the classic on-line problem of bin packing. It has long been observed that known bin packing algorithms with optimal average-case performance were not optimal in the worst-case sense. In particular First Fit and Best Fit had optimal average-case ratio of 1 but a worst-case competitive ratio of 1.7. The wasted space of First Fit and Best Fit for a uniform random sequence of length $n$ is expected to be $\Theta(n^{2/3})$ and $\Theta(\sqrt{n} \log ^{3/4} n)$, respectively. The competitive ratio can be improved to 1.691 using the Harmonic algorithm; further variations of this algorithm can push down the competitive ratio to 1.588. However, Harmonic and its variations have poor performance on average; in particular, Harmonic has average-case ratio of around 1.27. In this paper, first we introduce a simple algorithm which we term Harmonic Match. This algorithm performs as well as Best Fit on average, i.e., it has an average-case ratio of 1 and expected wasted space of $\Theta(\sqrt{n} \log ^{3/4} n)$. Moreover, the competitive ratio of the algorithm is as good as Harmonic, i.e., it converges to $1.691$ which is an improvement over 1.7 of Best Fit and First Fit. We also introduce a different algorithm, termed as Refined Harmonic Match, which achieves an improved competitive ratio of $1.636$ while maintaining the good average-case performance of Harmonic Match and Best Fit. Finally, our extensive experimental evaluation of the studied bin packing algorithms shows that our proposed algorithms have comparable average-case performance with Best Fit and First Fit, and this holds also for sequences that follow distributions other than the uniform distribution.

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