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 $\exists\mathbb R$-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].

翻译：我们设计了一种多项式时间算法，用于将简单多边形 $P$ 剖分成数量最少的星形多边形。该算法是否存在这一问题已公开四十余年 [Avis and Toussaint, Pattern Recognit., 1981]，并多次被提及，例如在 O'Rourke 的著名著作 [Art Gallery Theorems and Algorithms, 1987] 中。除了强烈的理论动机外，该问题还受到实际应用领域的推动，如数控铣削、运动规划和形状参数化。此前仅有的非平凡特例算法针对的是同时满足单调性和正交性的多边形 $P$ [Liu and Ntafos, Algorithmica, 1991]。对于一般多边形，仅已知禁止斯坦纳点的限制版本算法 [Keil, SIAM J. Comput., 1985]，即剖分中的每个拐角必须是 $P$ 的拐角。有趣的是，对于无限制版本解为常数规模的实例，限制版本的解规模可能是线性的。其覆盖变体（允许星形片重叠，即艺术画廊问题）近期被证明是 $\exists\mathbb R$-完全的，因此可能不属于 NP [Abrahamsen, Adamaszek and Miltzow, STOC 2018 & J. ACM 2022]；这与我们的结果形成鲜明对比。与我们的工作最相关的研究或许是 Chazelle 和 Dobkin 提出的将简单多边形剖分成最少凸片的多项式时间算法 [STOC, 1979 & Comp. Geom., 1985]。