The steiner problem with edge lengths 1 and 2,
Information Processing Letters
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
Approximation schemes for Euclidean k-medians and related problems
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
Journal of the ACM (JACM)
Improved Steiner tree approximation in graphs
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for constrained for constrained node weighted steiner tree problems
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Bypassing the embedding: algorithms for low dimensional metrics
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Primal-dual approximation algorithms for Node-Weighted Steiner Forest on planar graphs
Information and Computation
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In this paper, we consider the following variant of the geometric Steiner tree problem. Every point u which is not included in the tree costs a penalty of π(u) units. Furthermore, every Steiner point we use costs cS units. The goal is to minimize the total length of the tree plus the penalties. We prove that the problem admits a polynomial time approximation scheme, if the points lie in the plane. Our PTAS uses a new technique which allows us to bypass major requirements of Arora's framework for approximation schemes for geometric optimization problems [1]. It may thus open new possibilities to find approximation schemes for geometric optimization problems that have a complicated topology. Furthermore the techniques we use provide a more general framework which can be applied to geometric optimization problems with more complex objective functions.