Bounded Degree Group Steiner Tree Problems

TL;DR

This paper investigates bounded degree group Steiner tree problems, providing approximation algorithms and bounds for various graph classes and problem variants, advancing the understanding of degree-constrained Steiner tree optimization.

Contribution

It introduces approximation algorithms for bounded degree group Steiner tree problems, including new bounds for specific graph classes and problem variants, extending prior work to degree-constrained scenarios.

Findings

Approximation ratio $ ho imes O(\log k)$ for Steiner $k$-Tree based on Min-Degree Group Steiner Tree.

Achieves $O(\log^3 n)$ approximation for Min-Degree Group Steiner Tree on bounded treewidth graphs.

Provides bicriteria approximation for bounded degree group Steiner tree on trees, generalizing previous results.

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 ${\cal S}$ 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 $deg_T(v) \leq b(v)$ for all $v \in V$. We give a bicriteria $(O(\log N \log |{\cal S}|),O(\log^2 n))$-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.