Bounded Degree Group Steiner Tree Problems

Motivated by some open problems posed in [13], we study three problems that seek a low degree subtree T of a graph G =(V, E). In the Min-Degree Group Steiner Tree problem we are given a collection of node subsets (groups), and T should contain a node from every group. In the Min-Degree Steiner k-Tree problem we are given a set R of terminals and an integer k, and T should contain k terminals. In both problems the goal is to minimize the maximum degree of T . In the more general Degrees Bounded Min-Cost Group Steiner Tree problem, we are also given edge costs and individual degree bounds (Formula Presented). The output tree T should obey the degree constraints deg_T (v) ≤ b_v for all (Formula Presented), and among all such trees we seek one of minimum cost. When the input is a tree, an O(log² n) approximation for the cost is given in [10]. Our first result generalizes [10] – we give a bicriteria (O(log² n), O(log² n))-approximation algorithm for Degrees Bounded Min-Cost Group Steiner Tree problem on tree inputs. This matches the cost ratio of [10] but also approximates the degrees within O(log² n). Our second result shows that if Min-Degree Group Steiner Tree admits ratio ρ then Min-Degree Steiner k-Tree admits ratio ρ · O(log k). Combined with [12], this implies an O(log³ n)-approximation for Min-Degree Steiner k-Tree on general graphs, in quasi-polynomial time. Our third result is a polynomial time O(log³ n)-approximation algorithm for Min-Degree Group Steiner Tree on bounded treewidth graphs.