On Minimum Generalized Manhattan Connections

Abstract

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 $O(\mathrm {log}\;n)$-approximation is trivial. We show that the problem is NP-hard and present an $O(\sqrt{\mathrm {log}\;n})$-approximation algorithm. Moreover, we provide an $O(\mathrm {log}\;\mathrm {log}\;n)$-approximation algorithm for complete k-partite 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.

The full version of this paper [1] can be found at https://arxiv.org/abs/2010.14338.