Garey and Johnson [1979, pp. 124-127] show that the one-dimensional bin packing problem (1D-BPP) is NP-complete in the strong sense. Because the one-dimensional problem can be transformed into higher dimensional cases by adding extra, fixed dimensions, MD-BPP is also strongly NP-complete. The above transformation does not apply to some of the special cases of MD-BPP, and results concerning the complexity of these cases rely on other approaches. As an example, if the items or bins are square, then the transformation is not straightforward, because fixing one of the sizes would also fix the other, turning it into a trivial problem, at least for orthogonal MD-BPP.
Li and Cheng [1989] show that 2D-KP remains strongly NP-complete when packing squares in a rectangular bin, or packing rectangles into a square bin. Leung et al [1990] prove that packing squares inside a square bin also remains strongly NP-complete. But their proof, in which the 3-Partition Problem [e.g., Garey and Johnson 1979] is transformed into a 2D-KP, requires a large number of different size square items. Because the orthogonal problem with only one square size is trivial, and the general cases are strongly NP-complete, there must be a number of distinct sizes, greater than 1, where the problem becomes hard.
The transformation of 1D-BPP into MD-BPP helps to prove complexity results for the latter, but it does not directly offer insight into the real difficulties of solving multidimensional problems. In the 1D-BPP, the solution is given by partitioning the set of items into subsets, with each subset assigned to a bin. The only requirement is that the sum of the length of the items in each subset does not exceed a given bound, the capacity of the bin. Because addition is commutative, the order in which the items are packed is irrelevant. This is easily observed when considering the one-dimensional version of PLP. If the bin has length X and the item length l, then the number of items that can be packed is X l.
In the multi-dimensional case, even when the subsets are given, it is still necessary to verify the feasibility of packing the items. To verify feasibility, we must solve a MD- KP, with the value of each item given by its area. This MD-KP, alone, is NP-complete.
The solution of MD-BPP must include a partition of the items into subsets assigned to each bin, and the arrangement, or packing pattern, of these items inside each bin. The most common way to represent this arrangement is to assign each item to a position relative to some reference point in the bin. For example, each item can be assigned a pair of coordinates (xi, yi), corresponding to the horizontal and vertical distance to the reference point in the bin. If only normal patterns are considered, then the number of positions an item can take in the bin is finite, although usually large [Christofides and Whitlock 1977].
Another approach to represent the arrangement is using Sequence Pairs [Murata et al 1995], where two ordered lists represent the relative position between pairs of items. Each of these two lists corresponds to permutations of {1,..., }n , where n is the number of
packed items. There are four possible relative positions between items andi j : j is to the
right of i, i is to the right of j, j is above i, or i is above j.
Graphs can also represent packing patterns (e.g., Biró and Boros [1984] and Lins et al [2002]). The main problem with this approach is that different representations can lead to the same arrangement, as given by the coordinates of each item [Christofides and Hadjiconstantinou 1995]. This adds to the complexity of search algorithms, even when the number of solutions is small.
If we consider only orthogonal problems with normal packing patterns, the decisions to solve MD-BPP can be divided in four main types:
Partition: Which item goes in which bin? The number of possible assignments increases as 2I . This decision is always present in MD-BPP, and the complexity can only be reduced by adding constraints on combinations of items that can be packed together in a bin.
Order: After choosing a partition, an algorithm must select the order in which the items are packed, and the number of options goes up with the factorial of the number of items in the partition. In PLP, because there is only one type of item, there is no ordering decision.
Orientation: Each rectangular item can have two distinct orientations in two dimensions, or up to six orientations in three dimensions. Because each item can usually be rotated independently of the other, there are an exponential number of possibilities.
Relative Position: After deciding the ordering and the orientation, it is still necessary to select the position of each item being packed relative to all items previously packed in the same bin. The number of possible positions for placing an item may increase with the number of patterns that can be used with the items already packed, and the number of different patterns increases with the factorial of the number of items in the partition. The complexity of this positioning is simplified by rules like “left-most downward” (e.g., Christofides and Hadjiconstantinou [1995]).
Most heuristics and approximate algorithms in the literature reduce the size of the solution space by using a sorting rule to decide the ordering of items, fixing the orientation of the items, or packing items according to layers (e.g., Coffman et al [1980], Chung et al [1982], Li and Cheng [1990]).
Although exact algorithms for most variations of MD-BPP have exponential run time, they can significantly differ in terms of complexity because of the way the solutions are represented. For example, the algorithm for oriented 2D-SPP in Murata et al [1995], using sequence pairs, has complexity O n n( 8 ! )2 [Xu et al 1997], while the MIP analyzed by
Chen et al [1995], when applied to oriented 2D-SPP, has complexity O(2 )2n , where n is the number of items to pack.