Approximation algorithms for constrained for constrained node weighted steiner tree problems

We consider a class of optimization problems, where the input is an undirected graph with two weight functions defined for each node, namely the node's profit and its cost. The goal is to find a connected set of nodes of low cost and high profit. We present approximation algorithms for three natural optimization criteria that arise in this context, all of which are NP-hard. The budget problem asks for maximizing the profit of the set subject to a budget constraint on its cost. The quota problem requires minimizing the cost of the set subject to a quota constraint on its profit. Finally, the prize collecting problem calls for minimizing the cost of the set plus the profit (here interpreted as a penalty) of the complement set. For all three problems, our algorithms give an approximation guarantee of O(\log n), where n is the number of nodes. To the best of our knowledge, these are the first approximation results for the quota problem and for the prize collecting problem, both of which are at least as hard to approximate as set cover. For the budget problem, our results improve on a previous O(\log^2 n) result of Guha, Moss, Naor, and Schieber. Our methods involve new theorems relating tree packings to (node) cut conditions. We also show similar theorems (with better bounds) using edge cut conditions. These imply bounds for the analogous budget and quota problems with edge costs which are comparable to known (constant factor) bounds.

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

[2]  Santosh S. Vempala,et al.  Improved approximation guarantees for minimum-weight k-trees and prize-collecting salesmen , 1995, STOC '95.

[3]  Sanjeev Arora,et al.  A 2 + ɛ approximation algorithm for the k-MST problem , 2000, SODA '00.

[4]  Samir Khuller,et al.  The Budgeted Maximum Coverage Problem , 1999, Inf. Process. Lett..

[5]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[6]  Santosh S. Vempala,et al.  A constant-factor approximation for the k-MST problem in the plane , 1995, STOC '95.

[7]  Santosh S. Vempala,et al.  A constant-factor approximation algorithm for the k MST problem (extended abstract) , 1996, STOC '96.

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

[9]  Sudipto Guha,et al.  Efficient recovery from power outage (extended abstract) , 1999, STOC '99.

[10]  R. Ravi,et al.  A nearly best-possible approximation algorithm for node-weighted Steiner trees , 1993, IPCO.

[11]  Robert D. Carr,et al.  Randomized metarounding , 2002, Random Struct. Algorithms.

[12]  Sudipto Guha,et al.  Improved combinatorial algorithms for the facility location and k-median problems , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[13]  Vijay V. Vazirani,et al.  Primal-dual approximation algorithms for metric facility location and k-median problems , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[14]  Dorit S. Hochba,et al.  Approximation Algorithms for NP-Hard Problems , 1997, SIGA.

[15]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

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

[17]  Robert D. Carr,et al.  Randomized metarounding (extended abstract) , 2000, STOC '00.