## Section: New Results

Keywords : minimum spanning tree.

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

Participant : Ernst Althaus.

In
[15] , 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
con its edges, and an integer
k1 . The goal in the
*minimum-cost
k-hop spanning tree* (
) is to compute a spanning tree
Tin
Gof minimum total cost such that the longest root-leaf-path in the tree has at most
kedges.

Our main result is an algorithm that computes a tree of depth at most
kand total expected cost
O(log
n) 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, and Bartal. In particular, we show that the
problem can be solved exactly in polynomial time when the cost metric
cis induced by a so called
*hierarchically well-separated tree* .