In an application of map labelling to air-traffic control, labels should be placed with as few overlaps as possible since labels include important information about airplanes. Motivated by this application, de Berg and Gerrits (Comput. Geom. 2012) proposed a problem of maximizing the number of free labels (i.e. labels not intersecting with any other label) and developed approximation algorithms for their problem under various label-placement models. In this paper, we propose an alternative problem of minimizing a degree of overlap at a point. Specifically, the objective of this problem is to minimize the maximum of $\lambda \left(p\right)$ over $p\in {\mathbb{R}}^2$, where $\lambda \left(p\right)$ is defined as the sum of weights of labels that overlap with a point p. We develop a 4-approximation algorithm by LP-rounding under the 4-position model. We also investigate the case when labels are rectangles with bounded height/length ratios.

在将地图标签应用到空中交通管制中时，应放置尽可能少的重叠标签，因为标签包含有关飞机的重要信息。受此应用程序的激励，de Berg和Gerrits（计算机地理杂志，2012年）提出了一个最大化免费标签（即不与任何其他标签相交的标签）数量的问题，并针对各种标签放置模型下的问题开发了一种近似算法。在本文中，我们提出了一个最小化一个点的重叠程度的替代问题。具体来说，此问题的目的是最大程度地减少$\lambda （p）$ 超过 $p\in {\mathbb{[R}}^{\mathrm{2个}}$， 在哪里 $\lambda （p）$定义为与点p重叠的标签的权重之和。我们在4位置模型下通过LP舍入开发了4近似算法。我们还研究了标签是具有限制的高度/长度比的矩形的情况。