Bicriteria Approximation Tradeoff for the Node-Cost Budget Problem
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
Bicriteria approximation tradeoff for the node-cost budget problem
ACM Transactions on Algorithms (TALG)
Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Primal-dual approximation algorithms for node-weighted steiner forest on planar graphs
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Solving connected subgraph problems in wildlife conservation
CPAIOR'10 Proceedings of the 7th international conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems
Primal-dual approximation algorithms for Node-Weighted Steiner Forest on planar graphs
Information and Computation
Improved approximation algorithms for (budgeted) node-weighted steiner problems
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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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 the set cover. For the budget problem, our results improve on a previous $O(\log^2 n)$ result of Guha et al. 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.