The Dynamic Vertex Minimum Problem and Its Application to Clustering-Type Approximation Algorithms

The dynamic vertex minimum problem (DVMP) is to maintain the minimum cost edge in a graph that is subject to vertex additions and deletions. DVMP abstracts the clustering operation that is used in the primal-dual approximation scheme of Goemans and Williamson (GW). We present an algorithm for DVMP that immediately leads to the best-known time bounds for the GW approximation algorithm for problems that require a metric space. These bounds include time O(n) for the prize-collecting TSP and other direct applications of the GW algorithm (for n the number of vertices) as well as the best-known time bounds for approximating the k-MST and minimum latency problems, where the GW algorithm is used repeatedly as a subroutine. Although the improvement over previous time bounds is by only a sublogarithmic factor, our bound is asymptotically optimal in the dense case, and the data structures used are relatively simple.