When trees collide: an approximation algorithm for the generalized Steiner problem on networks
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
A General Approximation Technique for Constrained Forest Problems
SIAM Journal on Computing
A nearly best-possible approximation algorithm for node-weighted Steiner trees
Journal of Algorithms
Spanning Trees---Short or Small
SIAM Journal on Discrete Mathematics
Efficient recovery from power outage (extended abstract)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Sharing the cost of multicast transmissions
Journal of Computer and System Sciences - Special issue on Internet algorithms
Approximate k-MSTs and k-Steiner trees via the primal-dual method and Lagrangean relaxation
Mathematical Programming: Series A and B
Saving an epsilon: a 2-approximation for the k-MST problem in graphs
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Approximation Algorithms for Constrained Node Weighted Steiner Tree Problems
SIAM Journal on Computing
Maximising lifetime for fault-tolerant target coverage in sensor networks
Proceedings of the twenty-third annual ACM symposium on Parallelism in algorithms and architectures
Hi-index | 0.00 |
Moss and Rabani [13] 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(logn)-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(logn))-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 $\Omega(\frac{1}{\log n})$ of the optimal prize. We improve these results from at least two aspects. Our first main result is a primal-dual O(logh)-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 [11] and Guha et al. [9]) leads to a much simpler algorithm than that of Moss and Rabani [13] for PCST. Our second main contribution is for the Budgeted node-weighted Steiner tree problem, which is also an improvement to Moss and Rabani [13] and Guha et al. [9]. In the unrooted case, we improve upon an O(log2n)-approximation of [9], and present an O(logn)-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 [13] to a bicriteria approximation ratio of (1+ε, O(logn)/ε2) for any positive (possibly subconstant) ε. 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 [13].