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A Faster, Better Approximation Algorithm for the Minimum Latency Problem

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.


Source: Archer, Aaron - Algorithms and Optimization Group, AT&T Labs-Research
Williamson, David P. - Department of Computer Science, School of Operations Research and Industrial Engineering, Cornell University


Collections: Computer Technologies and Information Sciences; Mathematics