Faster approximation algorithms for the minimum latency problem

摘要

In this paper, we give a 9.28-approximation algorithm for the minimum latency problem that uses only O(n log n) 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 k-MST problem which is called as a black box. Their algorithm can achieve a performance guarantee of 10.77 while making O(n² log n) PCST calls (via a k-MST algorithm of Garg), or a performance guarantee of 7.18 + ε while using n^O(1/ε) PCST calls (via a k-MST algorithm of Arora and Karakostas). In order to match our approximation ratio (i.e. setting ε = 2.10), the latter version requires O(n⁵ log² n) PCST calls, so our running time bound is faster by a factor of Θ(n⁴ log n). Since PCST can be implemented to run in O(n²) time, the overall running time of our algorithm is O(n³ log n).The basic idea for our improvement is that we do not treat the k-MST algorithm as a black box. Thus we are able to take advantage of some situations in which the PCST subroutine delivers a k-MST with an improved performance guarantee.