A road map can be interpreted as a graph embedded in the plane, in which each vertex corresponds to a road junction and each edge to a particular road section. In this paper, we consider the computational cartographic problem to place non-overlapping road labels along the edges so that as many road sections as possible are identified by their name, i.e., covered by a label. We show that this is NP -hard in general, but the problem can be solved in $$O(n^3)$$ O ( n 3 ) time if the road map is an embedded tree with n vertices and constant maximum degree. This special case is not only of theoretical interest, but our algorithm in fact provides a very useful subroutine in exact or heuristic algorithms for labeling general road maps.

中文翻译：

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在路线图中放置标签：算法和复杂性

道路地图可以解释为嵌入平面的图形，其中每个顶点对应一个路口，每个边对应一个特定的路段。在本文中，我们考虑计算制图问题，沿边缘放置不重叠的道路标签，以便尽可能多的路段通过名称标识，即被标签覆盖。我们证明这通常是 NP -hard，但是如果路线图是具有 n 个顶点和恒定最大度数的嵌入树，则问题可以在 $$O(n^3)$$O (n 3 ) 时间内解决。这种特殊情况不仅具有理论意义，而且我们的算法实际上在精确或启发式算法中提供了一个非常有用的子程序，用于标记一般路线图。