Map labeling is the problem of placing labels at corresponding graphical features on a map. There are two main optimization problems: the label number maximization problem and the label size maximization problem. In general, both problems are NP-hard for static maps. Recently, the widespread use of several applications, such as personal mapping systems, has increased the importance of dynamic maps and the label number maximization problem for dynamic cases has been studied. In this paper, we consider the label size maximization problem for points on rotating maps. Our model is as follows. For each label, an anchor point is chosen inside the label or on its boundary. Each label is placed such that the anchor point coincides with the corresponding point on the map. Furthermore, while the map fully rotates from 0 to 2π, the labels are placed horizontally according to the angle of the map. Our problem consists of finding the maximum scale factor for the labels such that the labels do not intersect, and determining the placing of the anchor points. We describe an O(n log n)-time and O(n)-space algorithm for the case where each anchor point is inside the label. Moreover, if the anchor points are on the boundaries, we also present an O(n log n)-time and O(n)-space exact and approximation algorithms for several label shapes.

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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 λ(p) over p ∈ R 2 , where λ(p) 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.