🤖 AI Summary
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.
📝 Abstract
We devise a polynomial-time algorithm for partitioning a simple polygon P into a minimum number of star-shaped polygons. The question of whether such an algorithm exists has been open for more than four decades [Avis and Toussaint, Pattern Recognit., 1981] and it has been repeated frequently, for example in O’Rourke’s famous book [Art Gallery Theorems and Algorithms, 1987]. In addition to its strong theoretical motivation, the problem is also motivated by practical domains such as CNC pocket milling, motion planning, and shape parameterization. The only previously known algorithm for a non-trivial special case is for P being both monotone and rectilinear [Liu and Ntafos, Algorithmica, 1991]. For general polygons, an algorithm was only known for the restricted version in which Steiner points are disallowed [Keil, SIAM J. Comput., 1985], meaning that each corner of a piece in the partition must also be a corner of P. Interestingly, the solution size for the restricted version may be linear for instances where the unrestricted solution has constant size. The covering variant in which the pieces are star-shaped but allowed to overlap—known as the Art Gallery Problem—was recently shown to be ∃ℝ-complete and is thus likely not in NP [Abrahamsen, Adamaszek and Miltzow, STOC 2018 & J. ACM 2022]; this is in stark contrast to our result. Arguably the most related work to ours is the polynomial-time algorithm to partition a simple polygon into a minimum number of convex pieces by Chazelle and Dobkin [STOC, 1979 & Comp. Geom., 1985].
