Summary: A Faster, Better Approximation Algorithm for the
Minimum Latency Problem
Aaron Archer # Asaf Levin + David P. Williamson #
We give a 7.18approximation algorithm for the minimum latency problem that uses only
O(n log n) calls to the prizecollecting Steiner tree (PCST) subroutine of Goemans and Williamson.
This improves the previous best algorithms in both performance guarantee and running time.
A previous algorithm of Goemans and Kleinberg for the minimum latency problem requires an
approximation algorithm for the kMST problem which is called as a black box for each value
of k. Their algorithm can achieve a performance guarantee of 10.77 while making O(n 2 log n)
PCST calls (via a kMST algorithm of Garg), or a performance guarantee of 7.18+# while using
n O(1/#) PCST calls (via a kMST algorithm of Arora and Karakostas). In all cases, the running
time is dominated by the PCST calls. Since the PCST subroutine can be implemented to run
in O(n 2 ) time, the overall running time of our algorithm is O(n 3 log n).
The basic idea for our improvement is that we do not treat the kMST algorithm as a
black box. This allows us to take advantage of some special situations in which the PCST
subroutine delivers a kMST with a performance guarantee of 2. We are able to obtain the
same approximation ratio that would be given by Goemans and Kleinberg if we had access to
2approximate kMST's for all values of k, even though we have them only for some values of k
that we are not able to specify in advance.