[论文解读] Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line

José Correa, Laurent Feuilloley+1 more|arXiv (Cornell University)|Sep 25, 2013

Computational Geometry and Mesh Generation被引用 1

一句话总结

本文证明了在与对角线相交的轴对齐矩形族中寻找最大独立集（MIS）问题是NP完全的，提出了当矩形与对角线下方相交时加权MIS的O(n²)时间算法，并证明了MIS与最小击中集（MHS）之间的对偶间隙（即MIS与MHS的比值）介于2到4之间，改进了先前的界，从而在该设定下得到了加权MIS的2-近似算法。

ABSTRACT

Abstract. Finding a maximum independent set (MIS) of a given fam-ily of axis-parallel rectangles is a basic problem in computational geom-etry and combinatorics. This problem has attracted significant atten-tion since the sixties, when Wegner conjectured that the corresponding duality gap, i.e., the maximum possible ratio between the maximum independent set and the minimum hitting set (MHS), is bounded by a universal constant. An interesting special case, that may prove use-ful to tackling the general problem, is the diagonal-intersecting case, in which the given family of rectangles is intersected by a diagonal. Indeed, Chepoi and Felsner recently gave a factor 6 approximation algorithm for MHS in this setting, and showed that the duality gap is between 3/2 and 6. In this paper we improve upon these results. First we show that MIS in diagonal-intersecting families is NP-complete, providing one smallest subclass for which MIS is provably hard. Then, we derive an O(n2)-time algorithm for the maximum weight independent set when, in addition the rectangles intersect below the diagonal. This improves and extends a classic result of Lubiw, and amounts to obtain a 2-approximation algo-rithm for the maximum weight independent set of rectangles intersecting a diagonal. Finally, we prove that for diagonal-intersecting families the duality gap is between 2 and 4. The upper bound, which implies an approximation algorithm of the same factor, follows from a simple com-binatorial argument, while the lower bound represents the best known lower bound on the duality gap, even in the general case. An extended abstract of a preliminary version of this work appears in the proceedings

研究动机与目标

• 确定轴对齐矩形族与对角线相交时最大独立集（MIS）问题的计算复杂性。
• 设计一种高效算法，用于计算所有矩形与对角线下方相交时的最大权独立集。
• 收紧与对角线相交的矩形族中MIS与最小击中集（MHS）之间对偶间隙的界。
• 改进此几何设定下加权MIS的近似因子。

提出的方法

• 通过从已知的NP完全问题归约，证明与对角线相交的矩形族中MIS问题的NP完全性。
• 设计一种O(n²)时间的动态规划算法，用于在所有矩形与对角线下方相交的约束下求解加权MIS。
• 基于区间结构和嵌套性质的组合论证，建立对偶间隙上界为4。
• 构造一个紧致示例，证明对偶间隙至少为2，从而确立下界。
• 利用与对角线相交矩形的结构特性，将Lubiw的经典无权MIS结果推广。
• 结合算法结果与对偶间隙的界，得出该设定下加权MIS的2-近似算法。

实验结果

研究问题

• RQ1 与对角线相交的矩形族中，最大独立集问题是否为NP完全？
• RQ2 当所有矩形与对角线下方相交时，能否设计出高效算法求解最大权独立集？
• RQ3 与对角线相交的矩形族中，对偶间隙（MIS/MHS比值）的最紧上界是什么？
• RQ4 能否在此设定下将加权MIS的近似因子改进至优于先前结果？
• RQ5 此几何设定下，对偶间隙的最佳已知下界是多少？

主要发现

• 与对角线相交的轴对齐矩形族中，最大独立集问题为NP完全，识别出该上下文中已知最小的MIS NP完全子类。
• 提出一种O(n²)时间算法，用于计算所有矩形与对角线下方相交时的最大权独立集，扩展了Lubiw的结果。
• 证明与对角线相交的矩形族中，对偶间隙至多为4，从而得到加权MIS的2-近似算法。
• 证明对偶间隙至少为2，代表该比值在一般矩形情形下的最佳已知下界。
• 4的上界通过基于相交矩形结构分解的简单组合论证得出。
• 本文通过将对偶间隙的界从[3/2, 6]收紧至[2, 4]，改进了先前结果，并在该设定下提供了加权MIS的多项式时间2-近似算法。