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== Abstract ==

== Abstract ==

Consider a graph G=(V,E) with a weight value w(v) associated with each vertex v. A demand is a pair of vertices (s,t). A subgraph H satisfies the demand if s and t are connected in H. In the (offline) node-weighted Steiner forest problem, given a set of demands the goal is to find the minimum-weight subgraph H which satisfies all demands.

/*Consider a graph G=(V,E) with a weight value w(v) associated with each vertex v. A demand is a pair of vertices (s,t). A subgraph H satisfies the demand if s and t are connected in H. In the (offline) node-weighted Steiner forest problem, given a set of demands the goal is to find the minimum-weight subgraph H which satisfies all demands.

We give the first non-trivial approximation algorithm for the prize-collecting variant of the problem. In the prize-collecting variant, a penalty is associated with each demand. If the subgraph does not satisfy a demand, we need to pay the penalty of the demand. Our algorithm has a logarithmic approximation ratio which is tight up to a constant factor. Indeed our algorithm simplifies and generalizes the previous results for the prize-collecting node-weighted Steiner tree problem.

We give the first non-trivial approximation algorithm for the prize-collecting variant of the problem. In the prize-collecting variant, a penalty is associated with each demand. If the subgraph does not satisfy a demand, we need to pay the penalty of the demand. Our algorithm has a logarithmic approximation ratio which is tight up to a constant factor. Indeed our algorithm simplifies and generalizes the previous results for the prize-collecting node-weighted Steiner tree problem.