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 $P$ 划分为最小数量的星形多边形。关于此类算法是否存在的问题已悬而未决四十余年 [Avis and Toussaint, Pattern Recognit., 1981]，并在此后被频繁提及，例如在 O'Rourke 的著名著作 [Art Gallery Theorems and Algorithms, 1987] 中。除了具有强烈的理论动机外，该问题还受到实际应用领域的驱动，例如数控铣削、运动规划和形状参数化。此前已知的非平凡特例算法仅针对同时满足单调性和直线性的多边形 $P$ [Liu and Ntafos, Algorithmica, 1991]。对于一般多边形，仅在不允许使用 Steiner 点的限制版本中存在算法 [Keil, SIAM J. Comput., 1985]，这意味着划分中每个块的角点也必须是 $P$ 的角点。有趣的是，当无限制版本的解具有常数大小时，限制版本的解规模可能是线性的。允许星形块重叠的覆盖变体（即艺术画廊问题）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。与我们工作最相关的当属 Chazelle 和 Dobkin 提出的将简单多边形划分为最少数量凸块的多项式时间算法 [STOC, 1979 & Comp. Geom., 1985]。