Sona 绘制的新结果：困难性与 TSP 分离
New Results in Sona Drawing: Hardness and TSP Separation
AI总结：
本文证明最小长度 sona 绘制是 NP 困难的，且其长度可比 TSP 巡游长 1.5487875 倍以上，并证明网格限制下判定 sona 巡游是否存在也是 NP 困难的，回答了 CCCG 2006 的开放问题。
摘要：
给定一组点站点，sona 绘制是一条单一闭合曲线，该曲线与站点不相交，且仅在简单交叉处自交，使得其补集的每个有界区域恰好包含一个站点。我们证明，为给定的 n 个点寻找最小长度的 sona 绘制是 NP 困难的，并且这样的曲线可以比相同点的 TSP 巡游长出一个大于 1.5487875 的因子。当限制巡游位于正方形网格的边上、且点位于网格单元中时，我们证明即使判定这样的巡游是否存在也是 NP 困难的。这些结果回答了 CCCG 2006 上提出的问题。
英文摘要：
Given a set of point sites, a sona drawing is a single closed curve, disjoint from the sites and intersecting itself only in simple crossings, so that each bounded region of its complement contains exactly one of the sites. We prove that it is NP-hard to find a minimum-length sona drawing for $n$ given points, and that such a curve can be longer than the TSP tour of the same points by a factor $> 1.5487875$. When restricted to tours that lie on the edges of a square grid, with points in the grid cells, we prove that it is NP-hard even to decide whether such a tour exists. These results answer questions posed at CCCG 2006.
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