Integrality ratio for group Steiner trees and directed steiner trees

The natural relaxation for the Group Steiner Tree problem, as well as for its generalization, the Directed Steiner Tree problem, is a flow-based linear programming relaxation. We prove new lower bounds on the integrality ratio of this relaxation. For the Group Steiner Tree problem, we show the integrality ratio is Ω(log2 k), where k denotes the number of groups; this holds even for input graphs that are Hierarchically Well-Separated Trees, introduced by Bartal [Symp. Foundations of Computer Science, pp. 184–193, 1996], in which case this lower bound is tight. This also applies for the Directed Steiner Tree problem. In terms of the number n of vertices, our results for the Directed Steiner problem imply an Ω( log2 n (log log n)2 ) integrality ratio. For both problems, these are the first lower bounds on the integrality ratio that are superlogarithmic in the input size. This exhibits, for the first time, a relaxation of a natural optimization problem whose integrality ratio is ...

• Type: Conference
• Year: 2003
• Where: SODA
• Authors: Eran Halperin, Guy Kortsarz, Robert Krauthgamer, Aravind Srinivasan, Nan Wang
• Keywords: Algorithms, Integrality Ratio, SODA 2003, Steiner Tree Problem, Steiner Trees