## 1 Introduction  
An axial line is a geometrical element that represents a convex space, i.e., a spatial unit comprising the entire geometry of building and/or urban layout. An intersection between two axial lines thus indicates the link of the convex spaces represented by the axial lines. By using an axial map of a built environment, people can recognize the abstract spatial configuration of linear two-dimensional shapes [1; 2]. An axial line is defined as the longest line that can be drawn in a divided convex space that links to other axial lines that cover all of the given space without repetition [1]. An axial line is also considered to be a linear representation of a spatial configuration [2]. Thus, an axial map with the minimal set of axial lines characterizes a spatial, geometric and topological relationship [3]. As axial lines link the convex spaces within building interiors and urban layouts, these lines connect with other axial lines to show the spatial relationship of connectivity between convex spaces. These connections of axial lines as an axial map describe the geometries of a building or an urban space. Thus, this paper is based on the concept that a minimal set of axial lines is an axial map and that these axial lines represent the topological connectivity of geometric divisions in the built environment. However, many studiesstill manually create axial lines or choose generating all axial lines that hardly illustrate representative spatial connections. Conventional methods to select axial lines among all axial lines were based on greedy elimination algorithm and whether or not such selections are minimal is unclear [2].  
This paper develops an algorithm for finding a minimal set of axial lines that can represent a geometry of building and urban layout in two dimensions. [PERSON] et al. [2] stated that they had no proof that the axial lines selected by their greedy algorithm actually represented the minimum set of axial lines. As these researchers could not readily solve the set cover problem algorithmically, they used an approximated greedy algorithm. This method, however, could not promise to identify the real minimum of axial lines needed to represent an entire geometry of building and urban layout. If the minimum set of axial lines can be found, then an axial line map with the fewest and longest lines can be achieved that correctly guarantees the minimum set of axial lines in a building. Although [PERSON] et al. [2] noted the computational difficulty of the set cover problem, this paper seeks to overcome this difficulty and fully represent the geometry of built spaces.  
To represent a geometry of building and urban layout by axial lines, this paper requires a linear programming method for finding a global optimum and minimal set of axial lines. As a set of axial lines describes a spatial configuration of two-dimensional shapes, such sets of lines have been used for analyzing the spatial characteristics of buildings and urban places. Many studies still use manually drawn axial lines for spatial analyses and therefore many researchers have investigated methods for generating axial lines with computer programs (e.g., [2; 3; 4; 5; 6]). Computer programs provide the advantage of consistent results, high speed, large capacity and error immunity. In addition, some studies have adopted the approach of using all-line axial maps, because these maps are easy to generate with computer programs (e.g., [7; 8]). However, although an all-line axial map is easy to create, the redundancy of axial lines in these maps often renders direct use of the maps impractical or counter-intuitive. Thus, due to the strong demand for a scientific approach, it is important to select a set of axial lines from an all-line axial map.