线段排列中的最小单元连接和分离
Minimum cell connection and separation in line segment arrangements
摘要：
我们研究的复杂性，以下细胞的连接和分离问题的部分安排。给定平面中的一组直线段和诱导排列的不同单元中的两个点$a$和$b$：（i）计算需要移除的段的最小数目，使得存在连接$a$到$b$的路径，该路径不与任何剩余段相交;（ii）计算需要移除的段的最小数目，使得由剩余段引起的排列具有单个单元;（iii）计算需要保留的段的最小数目，使得连接$a$到$b$的任何路径与所保留的段中的一些相交。我们证明了问题（i）和（ii）是NP-困难的，并讨论了一些特殊的，易处理的情况。最值得注意的是，我们提供了一个线性时间算法的一个变种的问题（i）的路径连接$a$到$b$必须留在一个给定的多边形$P$内的孔的数量恒定，段包含在$P$，和端点的线段的边界上的$P$。对于问题（iii），我们提供了一个时间算法。
英文摘要：
We study the complexity of the following cell connection and separation problems in segment arrangements. Given a set of straight-line segments in the plane and two points $a$ and $b$ in different cells of the induced arrangement: (i) compute the minimum number of segments one needs to remove so that there is a path connecting $a$ to $b$ that does not intersect any of the remaining segments; (ii) compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell; (iii) compute the minimum number of segments one needs to retain so that any path connecting $a$ to $b$ intersects some of the retained segments. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a linear-time algorithm for a variant of problem (i) where the path connecting $a$ to $b$ must stay inside a given polygon $P$ with a constant number of holes, the segments are contained in $P$, and the endpoints of the segments are on the boundary of $P$. For problem (iii) we provide a cubic-time algorithm.
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