IMPROVED APPROXIMATION ALGORITHMS FOR (BUDGETED) NODE-WEIGHTED STEINER PROBLEMS.

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 node-weighted prize-collecting 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 ( 1 log n ) of the optimal prize. We improve these results from at least two aspects. Our rst main result is a primal-dual O(log h)-approximation algorithm for a more general problem, node-weighted prize-collecting 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 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 the work of Moss and Rabani. In the unrooted case, we improve upon an existing O(log2 n)-approximation by Guha et al., and present an O(log n)-approximation algorithm without any budget violation. For the rooted case, where a speci ed vertex has to appear in the solution tree, we improve the bicriteria result of Moss and Rabani to the bicriteria approximation ratio of (1 + ;O(log n)=2) for any positive (possibly subconstant) ∈. That is, for any permissible budget violation 1 + ∈, we present an algorithm achieving a trade o in the guarantee for the prize. Indeed, we show that this is almost tight for the natural linear-programming relaxation used by us as well as in the previous works. [ABSTRACT FROM AUTHOR]