Construction of Plat-form Models for the Problemof Packing Geometrical Objects

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In this study, we proposed "Multi-sphere Scheme" to efficiently pack given two- or three-dimensional objects in a compact space, designed all the components of the scheme, and investigated fundamental theory on geometrical packings and graph drawings. We designed an algorithm that can directly transform given triangle-mesh data into data for Multi-sphere Scheme based on a graph-theoretical analysis. We developed a 3D visual interface for Multi-sphere Scheme, by which we can easily check computational results in a visualized form. We greatly improved our solver for packing rectangles so that a long-standing open benchmark instance is solved for the first time by our new solver. As for the theory part, we found a 2D representation of triconnected graphs so that a useful triconnected decomposition can be easily obtained, and a characterization of the graphs of non-convex polytopes in a certain class.In particular, the latter result is the first such result since Steinitz' theorem, a characterization of the graphs of convex polytopes is found 80 years ago.