Boundary labeling is a technique used to label dense sets of feature points in an illustration. It involves placing labels along a rectangular boundary box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we consider grouping and ordering constraints for boundary labeling: Grouping constraints enforce that all labels in a group are placed consecutively on the boundary, and ordering constraints enforce a partial order over the labels. We show that finding an admissible labeling for labels of uniform size that can be placed on fixed candidate positions on two opposite sides of the boundary is NP-complete. Furthermore, we show that it is also weakly NP-hard to find an admissible labeling for non-uniform labels that can slide along one side of the boundary. However, we obtain polynomial-time algorithms in the one-sided setting for either fixed candidate positions or uniform-height labels. Finally, we experimentally confirm that our approach has also practical relevance.

翻译：边界标注是一种用于对插图中密集特征点集进行标注的技术。它涉及沿矩形边界框放置标签，并通过无交叉引线将每个标签与其对应的特征连接起来。尽管边界标注已得到充分研究，但标签上的语义约束尚未被深入探讨。本文考虑了边界标注的分组和排序约束：分组约束强制要求一组中的所有标签在边界上连续放置，而排序约束则强制要求标签之间存在偏序关系。研究表明，对于可放置在边界两侧固定候选位置上的均匀大小标签，找到可接受的标注方式是NP完全的。此外，对于可沿边界一侧滑动的非均匀标签，找到可接受的标注方式也是弱NP难的。然而，在单侧设置中，针对固定候选位置或均匀高度标签，我们获得了多项式时间算法。最后，实验证实了我们的方法也具有实际意义。