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$P$划分为最少数量的星形多边形。该算法是否存在的问题已悬置超过四十年[Avis and Toussaint, Pattern Recognit., 1981]，并曾被多次提及，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，该问题也受到如CNC型腔铣削、运动规划和形状参数化等实际领域的推动。先前已知的唯一处理非平凡特殊情况的算法是针对$P$同时是单调且直交的情形[Liu and Ntafos, Algorithmica, 1991]。对于一般多边形，已知算法仅适用于禁止使用Steiner点的受限版本[Keil, SIAM J. Comput., 1985]，即划分中每个子多边形的顶点也必须是$P$的顶点。有趣的是，对于无限制解为常数大小的实例，受限版本的解大小可能是线性的。覆盖变体（其中子多边形为星形但允许重叠——即著名的美术馆问题）最近被证明是$\exists\mathbb R$完全的，因此很可能不属于NP[Abrahamsen, Adamaszek and Miltzow, STOC 2018 & J. ACM 2022]；这与我们的结果形成鲜明对比。可以说与我们工作最相关的是Chazelle和Dobkin将简单多边形划分为最少数量凸多边形的多项式时间算法[STOC, 1979 & Comp. Geom., 1985]。