Faster approximation algorithms for the minimum latency problem

In this paper, we give a 9.28-approximation algorithm for the minimum latency problem that uses only <i>O</i>(<i>n</i> log <i>n</i>) calls to the prize-collecting Steiner tree (PCST) subroutine of Goemans and Williamson. A previous algorithm of Goemans and Kleinberg for the minimum latency problem requires an approximation algorithm for the <i>k</i>-MST problem which is called as a black box. Their algorithm can achieve a performance guarantee of 10.77 while making <i>O</i>(<i>n</i>² log <i>n</i>) PCST calls (via a <i>k</i>-MST algorithm of Garg), or a performance guarantee of 7.18 + ε while using <i>n</i>^{<i>O</i>(1/ε)} PCST calls (via a <i>k</i>-MST algorithm of Arora and Karakostas). In order to match our approximation ratio (i.e. setting ε = 2.10), the latter version requires <i>O</i>(<i>n</i>⁵ log² <i>n</i>) PCST calls, so our running time bound is faster by a factor of Θ(<i>n</i>⁴ log <i>n</i>). Since PCST can be implemented to run in <i>O</i>(<i>n</i>²) time, the overall running time of our algorithm is <i>O</i>(<i>n</i>³ log <i>n</i>).The basic idea for our improvement is that we do not treat the <i>k</i>-MST algorithm as a black box. Thus we are able to take advantage of some situations in which the PCST subroutine delivers a <i>k</i>-MST with an improved performance guarantee.