Nesting algorithm for rectangular-based, fixed-orientation polygons

Question

I'm looking for an algorithm that is closely related to the 2-dimensional nesting problem (also known as lay planning, bin packing, and the cutting stock problem).

The main differences between this and normal nesting are:

• A fixed number of elements to be placed
• The elements cannot rotate
• The elements must be placed in as small an area as possible, as opposed to placing as many pieces as possible in a fixed-size area.
• The elements are rectangular pieces with 0-4 fixed-orientation rectangular corner cutouts (“Utah-like”), as opposed to being randomly shaped

Input: n “Utah-like”, fixed-orientation polygons. n is in the range 1-20. The figures are not shaped to align perfectly with each other. Example figure set:

Desired output: A fixed-orientation nesting for a scaleable area with predefined proportions. The scaleable area should be scaled to fit the nested figures as snugly as possible, like so:

Note the area where the figures' rectangular boundaries overlap, preventing this from being a more trivial rectangle-packing problem.

I have perused several packing questions on SE (e.g. 1, 2, 3) as well as puzzle-solving questions (e.g. 1, 2, 3), and also outside Stack Exchange (1, 2, 3), but they don't describe this particular problem or don't include source code. Related GitHub repos: 1, 2, 3. Online solvers: SVGnest.com and Nestable.xyz.

Answer 1

I will describe how you can solve this with an ILP (integer linear programming) solver. For the size of problem you have, I expect it will work acceptably well.

Let's focus on just two of your shapes, say shape $S_i$ and $S_j$. Let $(x_i,y_i)$ denote the position of some fixed point of $S_i$ (say, the lower-left corner), and $(x_j,y_j)$ the position of some fixed point of $S_j$. Then it is possible to write down a formula $\Phi_{i,j}$ that captures the condition that $S_i,S_j$ do not overlap. This formula will have the form

$$(x_j-x_i \ge \alpha_1 \land y_j-y_i \ge \beta_1) \lor \cdots (x_j-x_i \ge \alpha_k \land y_j-y_i \ge \beta_k),$$

except that some $\ge$'s might be replaced with $\le$. Here the $\alpha,\beta$ values are constants that can be easily calculated from the shapes $S_i,S_j$. We can express $\Phi_{i,j}$ in a form suitable for use with an ILP solver using the methods in Express boolean logic operations in zero-one integer linear programming (ILP). In particular, we can encode $\Phi_{i,j}$ as the linear inequalities

$$\begin{align*} x_j-x_i &\ge \alpha_1 - C(1-t_1)\\ y_j-y_i &\ge \beta_1 - C(1-t_1)\\ &\vdots\\ x_j-x_i &\ge \alpha_k - C(1-t_k)\\ y_j-y_i &\ge \beta_k - C(1-t_k)\\ t_1 + \cdots + t_k &\ge 1 \end{align*}$$

except that whenever you replace $\ge$ with $\le$, you also replace $-C(1-t_u)$ with $+C(1-t_u)$. Here, $t_1,\dots,t_k$ are fresh new variables that are specific to $\Phi_{i,j}$, and they are constrained to be integers and constrained to be 0 or 1 (via the inequalities $0 \le t_1 \le 1$, etc.) Also $C$ is a large constant.

Finally, introduce a variable $z$, which represents the width of the containing rectangle. The containing rectangle's height will be $\gamma z$ where $\gamma$ is a known constant (the aspect ratio).

Add constraints to ensure that all shapes fit within the rectangle $[0,z] \times [0,\gamma z]$. In particular, for shape $S_i$, we obtain inequalities on $x_i,y_i$ (something like $0 \le x_i \le z-\text{width}(S_i)$, $0 \le y_i \le \gamma z - \text{height}(S_i)$ if $(x_i,y_i)$ is the lower-left corner of $S_i$, but the exact inequalities may vary depending on which fixed point of $S_i$ you chose).

Add the linear inequalities for each $\Phi_{i,j}$, for every pair $i,j$ with $i\ne j$.

The objective function is to minimize $z \ge 0$.

Finally, solve the resulting ILP instance obtained in this way. This should give you an optimal nesting.