An Improved Approximation Algorithm for the Minimum Latency Problem

We give a 7.18-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. 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 k-MST 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(n² log n) PCST calls (via a k-MST algorithm of Garg), or a performance guarantee of 7.18+e while using n^O(1/e) PCST calls (via a k-MST 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(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. This allows us to take advantage of some special situations in which the PCST subroutine delivers a k-MST 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 2-approximate k-MST’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.