Approximation Schemes for Independent Set and Sparse Subsets of Polygons
Approximation Schemes for Independent Set and Sparse Subsets of Polygons
复制标题
多边形独立集和稀疏子集的逼近方案
作者：
Anna Adamaszek;Sariel Har;Andreas Wiese
本文提出了一个从平面上给定的多边形集合计算最大权独立多边形集合的（1+ε）-近似算法，该算法具有拟多项式的运行时间.相比之下，该问题最著名的多项式时间算法的近似比为nε。令人惊讶的是，我们可以将算法扩展到计算给定多边形集合的最大基数子集的问题，这些多边形的相交图满足一些稀疏性条件。例如，我们证明了可以近似多边形的最大子集，使得该子集的相交图是平面的或不包含长度为4的圈（即，K2，2）。我们的算法依赖于一个递归的分区计划，其骨干是存在的平衡削减与小的复杂性，从一个小的总重量的最优解相交的多边形。对于大的平行轴矩形，我们提供了一个多项式时间（1 + ε）-近似的最大重量独立集。具体来说，我们考虑的问题，每个矩形有一个边缘的长度至少是一个恒定的分数的边界框的所有输入元素的相应边缘的长度。这是PTAS已知的最一般的情况，它需要一个新的和涉及的划分方案，这应该是独立的利益。
We present a (1+ε)-approximation algorithm with quasi-polynomial running time for computing a maximum weight independent set of polygons from a given set of polygons in the plane. Contrasting this, the best-known polynomial time algorithm for the problem has an approximation ratio of nε. Surprisingly, we can extend the algorithm to the problem of computing the maximum cardinality subset of the given set of polygons whose intersection graph fulfills some sparsity condition. For example, we show that one can approximate the maximum subset of polygons such that the intersection graph of the subset is planar or does not contain a cycle of length 4 (i.e., K2,2). Our algorithm relies on a recursive partitioning scheme, whose backbone is the existence of balanced cuts with small complexity that intersect polygons from the optimal solution of a small total weight. For the case of large axis-parallel rectangles, we provide a polynomial time (1 + ε)-approximation for the maximum weight independent set. Specifically, we consider the problem where each rectangle has one edge whose length is at least a constant fraction of the length of the corresponding edge of the bounding box of all the input elements. This is now the most general case for which a PTAS is known, and it requires a new and involved partitioning scheme, which should be of independent interest.