NP-Hardness of a special case of orthogonal packing problem

Question

Let $V$ be a set of $D$-dimensional rectangular shapes. For $d \in \{1,...,D\}$ and $v \in V$, $w_d(v) \in \mathbb{Q}^{+}$ describes the length of $v$ in the dimension $d$. The same notation is used for the container $C$. The $D$-dimensional orthogonal packing problem (OPP-$D$) is to decide if $V$ fits into the container $C$ without overlapping. Formally speaking, the problem is to find out whether $\forall d \in \{1,...,D\}$ there exists a function $f_d:V\rightarrow \mathbb{Q}^{+}$, such that $\forall v \in V, f_d(v)+w_d(v) \leq w_d(C)$ and $\forall v_1,v_2 \in V$, $(v_1 \neq v_2)$, $[f_d(v_1),f_d(v_1)+w_d(v_1)) \cap [f_d(v_2),f_d(v_2)+w_d(v_2)) = \emptyset$.

The problem is NP-complete (see Fekete SP, Schepers J. "On higher-dimensional packing I: Modeling". Technical Report 97–288, University at zu Köln, 1997). The problem is NP-complete even for $D=2$. I am wondering, whether the orthogonal packing problem for a bounded number of types (i.e. sizes in each dimension) of items is still NP-complete or not. Till now I found a result in some paper about NP-completeness of packing squares into a square (see JOSEPH Y-T. LEUNG, TOMMY W . TAM, AND C. S. WONG, "Packing squares into a square", Journal of Parallel and Distributed Computing, Volume 10 Issue 3, Nov. 1990) which is already a restriction but I still don't know what happens when the number of types of items is bounded.

Thank you for your answer,

Answer 1

I think that the paper by Klaus Jansen and Roberto Solis-Oba "An OPT + 1 Algorithm for the Cutting Stock Problem with Constant Number of Object Lengths" has a partial answer to your question. They consider a special case of your problem known as Cutting Stock problem when the number of different object types is constant and defined as follows:

In the cutting stock problem we are given a set $T = \{T_1,T_2,\dots,T_d\}$ of object types, where objects of type $T_i$ have positive integer length $p_i$. Given an infinite set of bins, each of integer capacity $\beta$, the problem is to pack a set $\mathcal{O}$ of $n$ objects into the minimum possible number of bins in such a way that the capacity of the bins is not exceeded; in set $\mathcal{O}$ there are $n_i$ objects of type $T_i$, for all $i =1,\dots,d$.

The authors claim that

it is not known whether the cutting stock problem can be solved in polynomial time for every fixed value $d$.

And they propose $OPT+1$ approximation polynomial-time algorithm when $d$ is fixed.

Since it's not proved that this special case is in $P$, this is the evidence that your problem is $NP$-hard.

Addendum: it's known that the case with two object types ($d=2$) is polynomially solvable, but for $d=3$ there is known only $OPT+1$-approximation.