Harmonic Algorithms for Packing d-Dimensional Cuboids into Bins.

We study harmonic-based algorithms for the $d$-dimensional ($d$D) generalizations of three classical geometric packing problems: geometric bin packing (BP), strip packing (SP), and geometric knapsack (KS). Caprara (MOR 2008) studied a harmonic-based algorithm $\mathtt{HDH}_k$, that has an asymptotic approximation ratio of $T_{\infty}^{d-1}$ (where $T_{\infty} \approx 1.691$) for $d$D BP and $d$D SP when items are not allowed to be rotated. We give fast and simple harmonic-based algorithms with asymptotic approximation ratios of $T_{\infty}^{d-1}$, $T_{\infty}^{d}$ and $(1-\epsilon)3^{-d}$ for $d$D SP, $d$D BP and $d$D KS, respectively, when orthogonal rotations are allowed about all or a subset of axes. This gives the first approximation algorithm for $d$D KS for $d > 3$. Furthermore, we provide a more sophisticated harmonic-based algorithm, which we call $\mathtt{HGaP}_k$, that is $T_{\infty}^{d-1}(1+\epsilon)$-asymptotic-approximate for $d$D BP for the rotational case. This gives an approximation ratio of $2.860 + \epsilon$ for 3D BP with rotations, which improves upon the current best-known algorithm. In addition, we study multiple-choice packing problems that generalize the rotational case. Here we are given $n$ sets of $d$D cuboidal items and we have to choose exactly one (resp. at most one for the knapsack variant) item from each set and then pack the chosen items. All our algorithms also work for multiple-choice packing problems.