We consider minimum-cardinality Manhattan connected sets with arbitrary demands: Given a collection of points $P$ in the plane, together with a subset of pairs of points in $P$ (which we call demands), find a minimum-cardinality superset of $P$ such that every demand pair is connected by a path whose length is the $\ell_1$-distance of the pair. This problem is a variant of three well-studied problems that have arisen in computational geometry, data structures, and network design: (i) It is a node-cost variant of the classical Manhattan network problem, (ii) it is an extension of the binary search tree problem to arbitrary demands, and (iii) it is a special case of the directed Steiner forest problem. Since the problem inherits basic structural properties from the context of binary search trees, an $O(\log n)$-approximation is trivial. We show that the problem is NP-hard and present an $O(\sqrt{\log n})$-approximation algorithm. Moreover, we provide an $O(\log\log n)$-approximation algorithm for complete bipartite demands as well as improved results for unit-disk demands and several generalizations. Our results crucially rely on a new lower bound on the optimal cost that could potentially be useful in the context of BSTs.

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关于最小广义曼哈顿连接

我们考虑具有任意需求的最小基数曼哈顿连接集：给定平面中的一组点 $P$，以及 $P$ 中点对的子集（我们称之为需求），找到一个最小基数超集$P$ 使得每个需求对都由长度为该对的 $\ell_1$-distance 的路径连接。这个问题是在计算几何、数据结构和网络设计中出现的三个经过充分研究的问题的变体：（i）它是经典曼哈顿网络问题的节点成本变体，（ii）它是任意要求的二叉搜索树问题，以及（iii）它是有向 Steiner 森林问题的特例。由于该问题从二叉搜索树的上下文中继承了基本结构属性，因此 $O(\log n)$ 近似值是微不足道的。我们证明该问题是 NP-hard 问题，并提出了一个 $O(\sqrt{\log n})$-近似算法。此外，我们为完整的二分需求提供了一个 $O(\log\log n)$-近似算法，以及对单元磁盘需求和几个概括的改进结果。我们的结果主要依赖于最优成本的新下限，该下限可能在 BST 的背景下有用。