Motivated by some open problems posed in , 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 degT (v) ≤ bv 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 . Our first result generalizes  – 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  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 , 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.
|Title of host publication||Combinatorial Algorithms - 31st International Workshop, IWOCA 2020, Proceedings|
|Editors||Leszek Gasieniec, Leszek Gasieniec, Ralf Klasing, Tomasz Radzik|
|Number of pages||12|
|State||Published - 2020|
|Event||31st International Workshop on Combinatorial Algorithms, IWOCA 2020 - Bordeaux, France|
Duration: 8 Jun 2020 → 10 Jun 2020
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Conference||31st International Workshop on Combinatorial Algorithms, IWOCA 2020|
|Period||8/06/20 → 10/06/20|
Bibliographical notePublisher Copyright:
© Springer Nature Switzerland AG 2020.