Moss and Rabani study constrained node-weighted Steiner tree problems with two independent weight values associated with each node, namely, cost and prize (or penalty). They give an O(log n)-approximation algorithm for the prize-collecting node-weighted Steiner tree problem (PCST)-where the goal is to minimize the cost of a tree plus the penalty of vertices not covered by the tree. They use the algorithm for PCST to obtain a bicriteria (2,O(log n))-approximation algorithm for the Budgeted node-weighted Steiner tree problem-where the goal is to maximize the prize of a tree with a given budget for its cost. Their solution may cost up to twice the budget, but collects a factor Ω(1/(log n)) of the optimal prize. We improve these results from at least two aspects. Our first main result is a primal-dual O(log h)-approximation algorithm for a more general problem, prize-collecting node-weighted Steiner forest (PCSF), where we have h demands each requesting the connectivity of a pair of vertices. Our algorithm can be seen as a greedy algorithm which reduces the number of demands by choosing a structure with minimum cost-to-reduction ratio. This natural style of argument (also used by Klein and Ravi and Guha et al.) leads to a much simpler algorithm than that of Moss and Rabani for PCST. Our second main contribution is for the Budgeted node-weighted Steiner tree problem, which is also an improvement to Moss and Rabani and Guha et al.. In the unrooted case, we improve upon an O(log~2 n)-approximation of, and present an O(log n)-approximation algorithm without any budget violation. For the rooted case, where a specified vertex has to appear in the solution tree, we improve the bicriteria result of to a bicriteria approximation ratio of (1 + ε, O(log n)/ε~2) for any positive (possibly subconstant) e. That is, for any permissible budget violation 1 + ε, we present an algorithm achieving a tradeoff in the guarantee for prize. Indeed, we show that this is almost tight for the natural linear-programming relaxation used by us as well as in.
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