Minimum Star Partitions of Simple Polygons in Polynomial Time
This paper resolves the long-standing “minimum star-shaped partition of a simple polygon” problem—open since 1981—by covering a given simple polygon with the fewest non-overlapping star-shaped subpolygons, allowing Steiner points. The proposed method integrates geometric decomposition, visibility graph optimization, dynamic programming, and structural analysis of star kernels, constructing the DP state space over triangulations. It yields the first exact polynomial-time algorithm applicable to arbitrary simple polygons, overcoming prior restrictions to monotone or orthogonal polygons and eliminating the requirement to forbid Steiner points. The algorithm runs in O(n⁹) time, a substantial improvement over exponential brute-force approaches. This theoretical breakthrough enables direct applications in CNC pocket milling, motion planning, and shape parameterization, where minimal star-shaped decompositions are essential for efficient toolpath generation, collision-free navigation, and domain mapping.