Computational Geometry ( IF 0.4 ) Pub Date : 2020-01-31 , DOI: 10.1016/j.comgeo.2020.101612 Adrian Dumitrescu
According to a result of Arkin et al. (2016), given n point pairs in the plane, there exists a simple polygonal cycle that separates the two points in each pair to different sides; moreover, a -factor approximation with respect to the minimum length can be computed in polynomial time. Here the following results are obtained:
(I) We extend the problem to geometric hypergraphs and obtain the following characterization of feasibility. Given a geometric hypergraph on points in the plane with hyperedges of size at least 2, there exists a simple polygonal cycle that separates each hyperedge if and only if the hypergraph is 2-colorable.
(II) We extend the -factor approximation in the length measure as follows: Given a geometric graph , a separating cycle (if it exists) can be computed in time, where , . Moreover, a -approximation of the shortest separating cycle can be found in polynomial time. Given a geometric graph in , a separating polyhedron (if it exists) can be found in time, where , . Moreover, a -approximation of a separating polyhedron of minimum perimeter can be found in polynomial time.
(III) Given a set of n point pairs in convex position in the plane, we show that a -approximation of a shortest separating cycle can be computed in time . In this regard, we prove a lemma on convex polygon approximation that is of independent interest.
中文翻译:
在最短的分离周期内
根据Arkin等人的结果。(2016年),给定平面中的n个点对,存在一个简单的多边形循环,将每对中的两个点分隔到不同的侧面;而且,一个可以在多项式时间内计算出与最小长度相关的系数近似值。在这里获得以下结果:
(I)将问题扩展到几何超图,并获得以下可行性描述。给定在平面上具有至少2个超边的点的几何超图,则存在一个简单的多边形循环,当且仅当超图是2色的时,才将每个超边分开。
(II)我们扩展 长度度量中的系数近似,如下所示:给定几何图 ,可以计算出一个分离周期(如果存在) 时间,地点 , 。此外,-最短分离周期的近似值可以在多项式时间内找到。给定几何图 在 ,可以在以下位置找到一个分离的多面体(如果存在) 时间,地点 , 。此外,最小周长的分离多面体的近似值可以在多项式时间内找到。
(III)给定一组n个点对在平面上的凸位置,我们证明-最短分离周期的近似值可以及时计算 。在这方面,我们证明了凸多边形逼近中的一个引人注目的引理。