Harmonic Algorithms for Packing
$d$-dimensional Cuboids Into Bins

For 2BP, Bansal et al. [3] obtained AAR of $T_{\infty}+\varepsilon$ even for the case with rotations, using a more sophisticated algorithm that used properties of harmonic rounding. Then there has been a series of improvements [3, 19] culminating with the present best AAR of 1.406 [5], for both the cases with and without orthogonal rotations. Bansal et al. [6] showed that $d$BP is APX-hard, even for $d=2$. They also gave an asymptotic PTAS for $d$BP where all items are $d$D squares.

Closely related to $d$BP is the $d$D strip packing problem ($d$SP), where we have to compute a packing of $I$ (without rotating the items) into a $d$D cuboid (called a strip) that has length one in the first $d-1$ dimensions and has the minimum possible length (called height) in the $d^{\textrm{th}}$ dimension.

For 2SP, an asymptotic PTAS was given by Kenyon and Rémila [22]. Jansen and van Stee [21] extended this to the case with orthogonal rotations. For 3SP, when rotations are not allowed, Bansal et al. [4] gave a harmonic-based algorithm achieving AAR of $T_{\infty}+\varepsilon$. Recently, this has been improved to $1.5+\varepsilon$ [20]. Miyazawa and Wakabayashi [25] studied 3SP and 3BP when rotations are allowed, and gave algorithms with AAR 2.64 and 4.89, respectively. Epstein and van Stee [13] gave an improved AAR of 2.25 and 4.5 for 3SP and 3BP with rotations, respectively. The $\operatorname{\mathtt{HDH}}_{k}$ algorithm also works for $d$SP and has an AAR of $T_{k}^{d-1}$. For online $d$BP, there are harmonic-based $T_{\infty}^{d}$-asymptotic-approximation algorithms [12, 11], which are optimal for $O(1)$ memory algorithms.

We will now define the $d$D multiple-choice bin packing problem ($d$MCBP). This generalizes $d$BP and captures the notion of orthogonal rotation of items. This perspective will be helpful in designing algorithms for the rotational case. In $d$MCBP, we’re given a set $\mathcal{I}=\{I_{1},I_{2},\ldots,I_{n}\}$, where for each $j$, $I_{j}$ is a set of items, henceforth called an itemset. We have to pick exactly one item from each itemset and pack those items into the minimum number of bins. See Fig. 1 for an example of 2MCBP.

We can model rotations using multiple-choice packing: Given a set $I$ of items, for each item $i\in I$, create an itemset $I_{i}$ that contains all allowed orientations of $i$. Then the optimal solution to $\mathcal{I}\coloneqq\{I_{i}:i\in I\}$ will tell us how to rotate and pack items in $I$.

Some algorithms for 2D bin packing with rotations assume that the bin is square [3, 19, 5]. This assumption holds without loss of generality when rotations are forbidden, because we can scale the items. But if rotations are allowed, this won’t work because items $i_{1}$ and $i_{2}$ that are rotations of each other may stop being rotations of each other after they are scaled. Multiple-choice packing algorithms can be used in this case. For each item $i\in I$, we will create an itemset $I_{i}$ that contains scaled orientations of $i$.

Multiple-choice packing problems have been studied before. Lawler gave an FPTAS for the multiple-choice knapsack problem [23]. Patt-Shamir and Rawitz gave an algorithm for multiple-choice vector bin packing having AAR $O(\log d)$ and a PTAS for multiple-choice vector knapsack [27]. Similar notions have been studied in the scheduling of malleable or moldable jobs [31, 18].

After the introduction of the harmonic algorithm for online 1BP by Lee and Lee [24], many variants have found widespread use in multidimensional packing problems (both offline and online) [8, 3, 4, 2, 12, 11, 17, 28, 29]. They are also simple, fast, and easy to implement. For example, among algorithms for 3SP, 2BP and 3BP with practical running time, harmonic-based algorithms provide the best AAR.

In our work, we extend harmonic-based algorithms to $d$MCBP. $d$MCBP subsumes the rotational case for geometric bin packing, and we believe $d$MCBP is an important natural generalization of geometric bin packing that may be of independent interest.

In Section 3, we describe ideas from $\operatorname{\mathtt{HDH}}_{k}$ [8] that help us devise harmonic-based algorithms for $d$MCBP. In Section 4, we show an $O(Nd+nd\log n)$-time algorithm for $d$MCBP, called $\operatorname{\mathtt{fullh}}_{k}$, having an AAR of $T_{k}^{d}$, where $n$ is the number of itemsets and $N$ is the total number of items across all the $n$ itemsets. $\operatorname{\mathtt{fullh}}_{k}$ is a fast and simple algorithm that works in two stages: In the first stage, we select the smallest item from each itemset (we will precisely define smallest in Section 4). In the second stage, we pack the selected items into bins using a variant of the $\operatorname{\mathtt{HDH}}_{k}$ algorithm.

In Section 5, we show an algorithm for $d$MCBP, called $\operatorname{\mathtt{HGaP}}_{k}$, having an AAR of $T_{k}^{d-1}(1+\varepsilon)$ and having a running time of $N^{O(1/\varepsilon^{2})}n^{(1/\varepsilon)^{O(1/\varepsilon)}}+O(Nd+nd\log n)$. For $d\geq 3$, this matches the present best AAR for the case where rotations are forbidden. Also, for large $k$, this gives an AAR of roughly $T_{\infty}^{2}\approx 2.860$ for 3D bin packing when orthogonal rotations are allowed, which is an improvement over the previous best AAR of $4.5$ [13], an improvement after fourteen years.

As harmonic algorithms are ubiquitous in bin packing, we expect our results will have applications in other related problems.

Our techniques can be extended to some other packing problems, like strip packing and geometric knapsack. In LABEL:sec:hdhk-sp, we define the $d$D multiple-choice strip packing problem ($d$MCSP) and extend Caprara’s $\operatorname{\mathtt{HDH}}_{k}$ algorithm [8] to $d$MCSP. The algorithm has AAR $T_{k}^{d-1}$ and runs in time $O(Nd+nd\log n)$, where $n$ is the number of itemsets and $N$ is the total number of items across all itemsets. In LABEL:sec:hdhks, we define the $d$D multiple-choice knapsack problem ($d$MCKS), and for any $0<\varepsilon<1$, we show an $O(Nd+N\log N+Nn/\varepsilon+nd\log n)$-time algorithm that is $(1-\varepsilon)3^{-d}$-approximate.

Let $\mathcal{I}$ be a set of itemsets. Define $\operatorname{flat}(\mathcal{I})$ to be the union of all itemsets in $\mathcal{I}$.

Let $K$ be a set of items that contains exactly one item from each itemset in $\mathcal{I}$. Formally, for each itemset $I\in\mathcal{I}$, $|K\cap I|=1$. Then $K$ is called an assortment of $\mathcal{I}$. Let $\Psi(\mathcal{I})$ denote the set of all assortments of $\mathcal{I}$. In $d$MCBP, given an input instance $\mathcal{I}$, we have to select an assortment $K\in\Psi(\mathcal{I})$ and output a bin packing of $K$, such that the number of bins used is minimized. Therefore, $\operatorname{opt}_{d\mathrm{MCBP}}(\mathcal{I})=\min_{K\in\Psi(\mathcal{I})}\operatorname{opt}_{d\mathrm{BP}}(K)$.

Fekete and Schepers [14] present a useful approach for obtaining lower bounds on the optimal solution to bin packing problems. Their approach is based on weighting functions.

Theorem 1 is proved in LABEL:sec:dff-trn.

To obtain a lower-bound on $\operatorname{opt}_{d\mathrm{BP}}(I)$ using Theorem 1, Caprara [8] defined a function $f_{k}$. For an integer constant $k\geq 3$, $f_{k}:[0,1]\mapsto[0,1]$ is defined as

$f_{k}$ was originally defined and studied by Lee and Lee [24] for their online algorithm for 1BP, except that they used $k/(k-1)$ instead of $k/(k-2)$. Define $\operatorname{type}_{k}:[0,1]\mapsto[k]$ as

Define $T_{k}$ to be the smallest positive constant such that $H_{k}(x)\coloneqq f_{k}(x)/T_{k}$ is a weighting function. We call $H_{k}$ the harmonic weighting function. We can efficiently compute $T_{k}$ as a function of $k$ using ideas from [24]. Table 1 lists the values of $T_{k}$ for the first few $k$. It can also be proven that $T_{k}$ is a decreasing function of $k$ and $T_{\infty}\coloneqq\lim_{k\to\infty}T_{k}\approx 1.6910302$.

For a $d$D cuboid $i$, define $f_{k}(i)$ to be the cuboid whose length is $f_{k}(\ell_{j}(i))$ in the $j^{\textrm{th}}$ dimension, for each $j\in[d]$. For a set $I$ of $d$D cuboids, let $f_{k}(I)\coloneqq\{f_{k}(i):i\in I\}$. Similarly define $H_{k}(i)$ and $H_{k}(I)$. Define $\operatorname{type}(i)$ to be a $d$-dimensional vector whose $j^{\textrm{th}}$ component is $\operatorname{type}_{k}(\ell_{j}(i))$. Note that there can be at most $k^{d}$ different values of $\operatorname{type}(i)$. Sometimes, for the sake of convenience, we may express $\operatorname{type}(i)$ as an integer in $[k^{d}]$.

From the $\operatorname{\mathtt{HDH}}_{k}$ algorithm by Caprara [8], we extracted out a useful subroutine, which we call $\operatorname{\mathtt{HDH-unit-pack}}_{k}$, that satisfies the following useful property:

The design of $\operatorname{\mathtt{HDH-unit-pack}}_{k}$ and its correctness can be inferred from Lemma 4.1 in [8]. We use $\operatorname{\mathtt{HDH-unit-pack}}_{k}$ as a black-box subroutine in our algorithms, i.e., we only rely on Property 2; we don’t need to know anything else about $\operatorname{\mathtt{HDH-unit-pack}}_{k}$. Nevertheless, for the sake of completeness, in LABEL:sec:hdhkunit, we give a complete description of $\operatorname{\mathtt{HDH-unit-pack}}_{k}$ and prove its correctness.

As mentioned before, we can solve the rotational version of $d$BP by reducing it to $d$MCBP. Specifically, for each item $i$ in the $d$BP instance, we create an itemset containing all orientations of $i$, and we pack the resulting $d$MCBP instance using $\operatorname{\mathtt{fullh}}_{k}$. Since an item can have up to $d!$ allowed orientations, this can take up to $O(nd!+nd\log n)$ time. Hence, the running time is large when $d$ is large. However, we can do better for some special cases.

When the bin has the same length in each dimension, then for any item $i$, $\operatorname{vol}(f_{k}(i))$ is independent of how we orient $i$. Hence, we can orient the items $I$ arbitrarily and then pack them using $\operatorname{\mathtt{fullh}}_{k}$ in $O(nd\log n)$ time.

Suppose there are no orientation constraints, i.e., all $d!$ orientations of each item are allowed. Let $L_{j}$ be the length of the bin in the $j^{\textrm{th}}$ dimension, for each $j\in[d]$. To use $\operatorname{\mathtt{fullh}}_{k}$ to pack $I$, we need to find the best orientation for each item $i\in I$, i.e., we need to find a permutation $\pi$ for each item $i$ such that $\prod_{j=1}^{d}f_{k}\left(\ell_{\pi_{j}}(i)/L_{j}\right)$ is minimized. This can be formulated as a maximum-weight bipartite matching problem on a graph with $d$ vertices in each partition: for every $u\in[d]$ and $v\in[d]$, the edge $(u,v)$ has a non-negative weight of $-\log(f_{k}(\ell_{u}(i)/L_{v}))$. So, using the Kuhn-Munkres algorithm [26], we can find the best orientation for each item in $O(d^{3})$ time. Hence, we can pack $I$ using $\operatorname{\mathtt{fullh}}_{k}$ in $O(nd^{3}+nd\log n)$ time.

$\operatorname{\mathtt{HGaP}}_{k}$ relies on the following key structural theorem. We formally prove it in Section A.2 and give an outline of the proof here.

LABEL:sec:hgap:improve-time gives hints on improving the running time of $\operatorname{\mathtt{HGaP}}_{k}$.

We can solve the rotational version of $d$BP by reducing it to $d$MCBP and using the $\operatorname{\mathtt{HGaP}}_{k}$ algorithm. Since each item can have up to $d!$ orientations, the running time is polynomial in $nd!$, which is large when $d$ is large. But we can do better for some special cases.

When the bin has the same length in each dimension, then for any item $i$, $w(i)\coloneqq\prod_{j=1}^{d-1}f_{k}(\ell_{j}(i))$ is invariant to permuting the first $d-1$ dimensions. In the first step of $\operatorname{\mathtt{HGaP}}_{k}$, we replace each $d$D item $i$ by a rectangle of width $w(i)$ and height $\ell_{d}(i)$. So, instead of considering all $d!$ orientations, we just need to consider at most $d$ different orientations, where each orientation has a different length in the $d^{\textrm{th}}$ dimension.

Suppose there are no orientation constraints, i.e., all $d!$ orientations of each item are allowed. Let $L_{j}$ be the length of the bin in the $j^{\textrm{th}}$ dimension, for each $j\in[d]$. Analogous to the trick in Section 4.1, we first fix the $d^{\textrm{th}}$ dimension of the item and then optimally permute the first $d-1$ dimensions using a max-weight bipartite matching algorithm. Hence, we need to consider only $d$ orientations instead of $d!$.