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

The dynamic vertexminim um problem (DVMP) is to maintain the minimum cost edge in a graph that is subject to vertexadditions 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(n2) 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.

[1]  David P. Williamson,et al.  A general approximation technique for constrained forest problems , 1992, SODA '92.

[2]  Santosh S. Vempala,et al.  A Constant-Factor Approximation Algorithm for the k-MST Problem , 1999, J. Comput. Syst. Sci..

[3]  David P. Williamson,et al.  A primal-dual approximation algorithm for generalized steiner network problems , 1995, Comb..

[4]  Philip N. Klein,et al.  A Data Structure for Bicategories, with Application to Speeding up an Approximation Algorithm , 1994, Inf. Process. Lett..

[5]  R. Tarjan.  Amortized Computational Complexity , 1985 .

[6]  H. Gabow.  An E cient Approximation Algorithmfor the Survivable Network Design Problem , 1993 .

[7]  David P. Williamson,et al.  An efficient approximation algorithm for the survivable network design problem , 1998, Math. Program..

[8]  David P. Williamson,et al.  Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees, with Applications to Matching and Set Cover , 1993, ICALP.

[9]  Robert Sedgewick,et al.  Algorithms in C , 1990 .

[10]  Jon M. Kleinberg,et al.  An improved approximation ratio for the minimum latency problem , 1996, SODA '96.

[11]  Sunil Arya,et al.  A 2.5-Factor Approximation Algorithm for the k-MST Problem , 1998, Inf. Process. Lett..

[12]  Chuan Yi Tang,et al.  A 2.|E|-Bit Distributed Algorithm for the Directed Euler Trail Problem , 1993, Inf. Process. Lett..

[13]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[14]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[15]  Madhu Sudan,et al.  The minimum latency problem , 1994, STOC '94.

[16]  David S. Johnson,et al.  The prize collecting Steiner tree problem: theory and practice , 2000, SODA '00.

[17]  Naveen Garg,et al.  A 3-approximation for the minimum tree spanning k vertices , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[18]  Moshe Lewenstein,et al.  A faster implementation of the Goemans-Williamson clustering algorithm , 2001, SODA '01.

[19]  Andrew V. Goldberg,et al.  Improved approximation algorithms for network design problems , 1994, SODA '94.

[20]  R. Ravi,et al.  When Trees Collide: An Approximation Algorithm for the Generalized Steiner Problem on Networks , 1995, SIAM J. Comput..

[21]  R. Ravi,et al.  When trees collide: an approximation algorithm for the generalized Steiner problem on networks , 1991, STOC '91.