Bounded Degree Group Steiner Tree Problems

Abstract

We study two problems that seek a subtree $T$ of a graph $G=(V,E)$ such that $T$ satisfies a certain property and has minimal maximum degree. - In the Min-Degree Group Steiner Tree problem we are given a collection $\mathcal{S}$ of groups (subsets of $V$) 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 at least $k$ terminals. We show that if the former problem admits approximation ratio $\rho$ then the later problem admits approximation ratio $\rho \cdot O(\log k)$. For bounded treewidth graphs, we obtain approximation ratio $O({\log }^{3}n)$ for Min-Degree Group Steiner Tree. In the more general Bounded Degree Group Steiner Tree problem we are also given edge costs and degree bounds $\{b(v):v\in V\}$, and $T$ should obey the degree constraints $de{g}_T(v)\le b(v)$ for all $v\in V$. We give a bicriteria $(O(\log N\log |\mathcal{S}|),O({\log }^{2}n))$-approximation algorithm for this problem on tree inputs, where $N$ is the size of the largest group, generalizing the approximation of Garg, Konjevod, and Ravi for the case without degree bounds.