## 4 Problem Formulation  
A minimum set of axial lines can be found when every axial line has at least one intersection with the extension lines from the wall edges to the sides of the reflex angles among \(n\) axial lines and \(m\) extension lines. This assumption follows a concept proposed by [PERSON] et al., that such a set of axial lines represents the entire building's geometry by including all spaces and \"getting everywhere\" in the building [2]. According to the definitions used by [PERSON] et al. [2], s-lines (or \"surface lines\") are extension lines of the wall edges to the sides of reflex angles, and m-lines (or \"movement lines\") are diagonals of straight lines between the vertexes, excluding lines that intersect walls in a given space. [PERSON] et al. proposed that to include all of the geometry (or \"get everywhere\"), it is necessary that the m-lines cross every s-line, and that this set of m-lines becomes a set of axial lines that represent a building's geometry [2]. If every s-line has at least one intersection with \(n\) axial lines and \(m\) s-lines, then a set of axial lines can represent an entire building's geometry by including all spaces (or getting everywhere). This circumstance indicates that if a solution to the linear programming problem exists, then it will be guaranteed to be an optimum solution. The minimal set of axial lines calculated using this formulation will have intersections with every s-line. This special form of linear programming, namely zero-one integer linear programming, is adopted in this paper. This paper's method assumes that there are \(n\) axial lines and \(m\) s-lines in a given building plan. Let \(x_{i}=1\) when the \(i\)th axial line is selected (or placed in a reduced set), and let \(x_{i}=0\) otherwise, because the number of axial lines is to be minimized and the objective function of the problem is to minimize the sum of \(n\) variables (\(\min\sum_{i=1}^{n}x_{i}\)). For a formulation of the optimization problem, this paper uses the formation of intersection with a set of \(n\) axial lines. Intersection with a set of \(n\) axial lines is equivalent to intersection with at least one axial line in the set. To check whether an s-line has such an intersection, Equation (1) is used. \[a_{1}+a_{2}+\cdots+a_{n-1}+a_{n}\geq 1 \tag{1}\]  
where \(a_{i}=1\) if an s-line has an intersection with the \(i\)th of \(n\) axial lines, and 0 otherwise. In addition, the equation for an intersection with a subset is as follows:  
\[a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n-1}x_{n-1}+a_{n}x_{n}\ \geq 0 \tag{2}\]  
where \(x_{i}=1\) if an element is of the subset, and 0 otherwise. Suppose that an s-line has intersections with a subset of axial lines. That Equation (2) is greater than or equal to 1 is the necessary and sufficient condition for this s-line to intersect at least one axial line in the reduced subset. These equations indicate a way to formulate the optimization problem in this paper. Accordingly, Formulation 2 of the optimization problem can be stated as follows:  
minimize \(\sum_{i=1}^{n}x_{i}\),  
subject to  
\(\mathbf{AX}\geq\mathbf{B}\),  
where  
\(\mathbf{A}=\left[ \begin{array}{cccc}a_{1,1}&a_{1,2}&\cdots&a_{1,n-1}&a_{1,n} \\ a_{2,1}&a_{2,2}&&&a_{2,n}\\ \vdots&&\ddots&&\vdots\\ a_{m-1,1}&&&a_{m-1,n-1}&a_{m-1,n}\\ a_{m,1}&a_{m,2}&\cdots&a_{m,n-1}&a_{m,n}\\ \end{array} \right]\mathbf{X}=\left[ \begin{array}{c}x_{1}\\ x_{2}\\ \vdots\\ x_{n-1}\\ x_{n}\\ \end{array} \right]\mathbf{B}=\left[ \begin{array}{c}b_{1}\\ b_{2}\\ \vdots\\ b_{m-1}\\ b_{m}\\ \end{array} \right],\)  
\(a_{ij}=1\) if the \(i\)th s-line has an intersection with the \(j\)th axial line, and 0 otherwise,  
\(b_{i}=1\),\(x_{i}\in\ \{0,1\}\). **Formulation 2. ** Linear programming problem to find a minimal set of axial lines  
This approach involves a special form of linear programming, namely zero-one integer linear programming. If a solution to this problem exists, then the solution will be guaranteed to be an optimum solution.