Approximating k-hop minimum spanning trees in Euclidean metrics

Abstract

In the minimum-cost k-hop spanning tree (k-hop MST) problem, we are given a set S of n points in a metric space, a positive small integer k and a root point r ∈ S. We are interested in computing a rooted spanning tree of minimum cost such that the longest root-leaf path in the tree has at most k edges. We present a polynomial-time approximation scheme for the plane. Our algorithm is based on Arora's et al. [5] technique for the Euclidean k-median problem.

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

“…In these problems we are given a metric M and a positive integer h. The objective in the BDMST problem is to minimize the weight of a spanning tree T of M with hop-diameter at most h. In the closely related h-hop MST problem the objective is to minimize the weight of a rooted spanning tree with hop-radius at most h. Both problems are among the classical and most well-studied NP-hard problems. (See the book of Garey and Johnson [34], page 206, and [41,18,5,39,42,35].) Kortsarz and Peleg [41] and Charikar et al [18] devised an O(log n)-approximation algorithm for these problems when h is a constant, and an O(n ǫ )-approximation algorithm for an arbitrarily small ǫ > 0, that is applicable for all values of h. (These algorithms provide the same approximation guarantees for significantly more general problems.)…”

“…Kantor and Peleg [39] devised 2 O(h) -approximation algorithms for these problems. Laue and Matijevic [42] presented a PTAS for 2-dimensional Euclidean metrics when h is a constant. These problems were also studied empirically.…”

A PTAS for $k$-hop MST on the Euclidean plane: Improving Dependency on $k$

“…For any > 0, Laue and Matijević [14] present a polynomial-time approximation scheme (PTAS) for this problem on the plane that runs in time n O(k/ ) for finding a (1 + )-approximate solution. They follow the general framework of random dissection by Arora [3,4] for finding good approximate solutions for instances in Euclidean metric.…”

“…They follow the general framework of random dissection by Arora [3,4] for finding good approximate solutions for instances in Euclidean metric. The dynamic programming structure of [14] (reviewed in Section 3.1) follows the approach from the PTAS for the k-median problem by Arora, Raghavan, and Rao [5].…”

“…We review the PTAS framework in Section 2 and review the dynamic programming approach of [14] in Section 3.1. Section 3.2 discusses how we change the dynamic programming table to reduce the dependency on k. We give full descriptions of the dynamic programming in Section 3.4 and its analysis in Section 4.…”