## Project-Team : modbio

## Section: New Results

**Keywords: ***minimum spanning tree*.

## Approximating k-hop minimum spanning trees (ATIPE)

**Participant:**Ernst Althaus.

In a paper to appear in Operations Research Letters in 2005,
we consider the problem of computing minimum-cost
spanning trees with depth restrictions. Specifically, we are given an
n-node complete graph G, a metric cost-function c on its edges,
and an integer k1. The goal in the *minimum-cost k-hop
spanning tree* () problem is to compute a spanning tree T in G of
minimum total cost such that the longest root-leaf-path in the tree
has at most k edges.

Our main result is an algorithm that computes a tree of depth at most
k and total expected cost O(logn) times that of a minimum-cost
k-hop spanning-tree. The result is based upon earlier work on
metric space approximation due to Fakcharoenphol et al. [35],
and Bartal [27][28].
In particular, we show that the problem can be solved exactly
in polynomial time when the cost metric c is induced by a so called
*hierarchically well-separated tree*.