TY - GEN

T1 - Bounded Degree Group Steiner Tree Problems

AU - Kortsarz, Guy

AU - Nutov, Zeev

N1 - Publisher Copyright: © Springer Nature Switzerland AG 2020.

PY - 2020

Y1 - 2020

N2 - 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 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 [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.

AB - 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 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 [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.

UR - http://www.scopus.com/inward/record.url?scp=85086261339&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/978-3-030-48966-3_26

DO - https://doi.org/10.1007/978-3-030-48966-3_26

M3 - منشور من مؤتمر

SN - 9783030489656

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 343

EP - 354

BT - Combinatorial Algorithms - 31st International Workshop, IWOCA 2020, Proceedings

A2 - Gasieniec, Leszek

A2 - Klasing, Ralf

A2 - Radzik, Tomasz

T2 - 31st International Workshop on Combinatorial Algorithms, IWOCA 2020

Y2 - 8 June 2020 through 10 June 2020

ER -