Abstract
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 2) 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.
This work was supported in part by Texas Advanced Research Program Grant 003658-0029-1999, NSF Grant CCR-9988160 and an MCD Graduate Fellowship.
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Gabow, H.N., Pettie, S. (2002). The Dynamic Vertex Minimum Problem and Its Application to Clustering-Type Approximation Algorithms. In: Penttonen, M., Schmidt, E.M. (eds) Algorithm Theory — SWAT 2002. SWAT 2002. Lecture Notes in Computer Science, vol 2368. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45471-3_20
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DOI: https://doi.org/10.1007/3-540-45471-3_20
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