In this paper we study the prize-collecting version of the Generalized Steiner Tree problem. To the best of our knowledge, there is no general combinatorial technique in approximation algorithms developed to study the prize-collecting versions of various problems. These problems are studied on a case by case basis by Bienstock et al. [5] by applying an LP-rounding technique which is not a combinatorial approach. The main contribution of this paper is to introduce a general combinatorial approach towards solving these problems through novel primal-dual schema (without any need to solve an LP). We fuse the primal-dual schema with Farkas lemma to obtain a combinatorial 3-approximation algorithm for the PrizeCollecting Generalized Steiner Tree problem. Our work also inspires a combinatorial algorithm [19] for solving a special case of Kelly's problem [22] of pricing edges. We also consider the k-forest problem, a generalization of k-MST and k-Steiner tree, and we show that in spite of these problems for which there are constant factor approximation algorithms, the k-forest problem is much harder to approximate. In particular, obtaining an approximation factor better than O(n 1/6−ε ) for kforest requires substantially new ideas including improving the approximation factor O(n 1/3−ε ) for the notorious densest k-subgraph problem. We note that k-forest and prize-collecting version of Generalized Steiner Tree are closely related to each other, since the latter is the Lagrangian relaxation of the former.
Moreover, the approximability of these two versions of dense-k-subgraph are polynomially related REF .
The prize-collecting generalized steiner tree problem via a new approach of primal-dual schema
SODA '06
Computer Science