TY - JOUR
T1 - Approximation algorithms for constrained node weighted steiner tree problems
AU - Moss, A.
AU - Rabani, Y.
PY - 2007
Y1 - 2007
N2 - 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(log2 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.
AB - 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(log2 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.
KW - Combinatorial approximation algorithms
KW - Network design
KW - Node-weighted problems
UR - http://www.scopus.com/inward/record.url?scp=42549130433&partnerID=8YFLogxK
U2 - 10.1137/S0097539702420474
DO - 10.1137/S0097539702420474
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AN - SCOPUS:42549130433
SN - 0097-5397
VL - 37
SP - 460
EP - 481
JO - SIAM Journal on Computing
JF - SIAM Journal on Computing
IS - 2
ER -