Approximation algorithms for constrained node weighted steiner tree problems

Anna Moss*, Yuval Rabani

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

20 Scopus citations

Abstract

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 t o approximate as set cover. For the budget problem, our results improve on a previous O(log2 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.

Original languageAmerican English
Pages (from-to)373-382
Number of pages10
JournalConference Proceedings of the Annual ACM Symposium on Theory of Computing
StatePublished - 2001
Externally publishedYes
Event33rd Annual ACM Symposium on Theory of Computing - Creta, Greece
Duration: 6 Jul 20018 Jul 2001

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