On Minimum Generalized Manhattan Connections

TL;DR

This paper studies a generalized Manhattan network problem with arbitrary demands, proving NP-hardness and providing approximation algorithms with bounds improving over trivial solutions, relevant to computational geometry and network design.

Contribution

The paper introduces a new approximation algorithm for the minimum generalized Manhattan connection problem and establishes its NP-hardness, along with specialized bounds for certain demand types.

Findings

NP-hardness of the problem.

An $O(rac{1}{ oot})$-approximation algorithm.

An $O(rac{1}{ oot oot})$-approximation for bipartite demands.

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 $O(\log n)$-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.