Integrality ratio for group Steiner trees and directed steiner trees

We present an Ω(log2k) lower bound on the integrality ratio of the flow-based relaxation for the Group Steiner Tree problem, 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 relaxation appears to be the only one that have been studied for the problem, as well as for its generalization, the Directed Steiner Tree problem. For the latter problem, our results imply an Ω(log2n/(log logn)2) integrality ratio, where n is the number of vertices in the graph. For both problems, this is the first known lower bound on the integrality ratio that is superlogarithmic in the input size. We also show algorithmically that the integrality ratio for Group Steiner Tree is much better for certain families of instances, which helps pinpoint the types of instances that appear to be most difficult to approximate.

[1]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1994, JACM.

[2]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[3]  Yair Bartal,et al.  Probabilistic approximation of metric spaces and its algorithmic applications , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[4]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[5]  U. Feige.  A threshold of ln n for approximating set cover , 1998, JACM.

[6]  R. Ravi,et al.  A polylogarithmic approximation algorithm for the group Steiner tree problem , 2000, SODA '98.

[7]  Sudipto Guha,et al.  Approximation algorithms for directed Steiner problems , 1999, SODA '98.

[8]  Sudipto Guha,et al.  Rounding via Trees : Deterministic Approximation Algorithms forGroup , 1998 .

[9]  Yair Bartal,et al.  On approximating arbitrary metrices by tree metrics , 1998, STOC '98.

[10]  Béla Bollobás,et al.  A Ramsey-type theorem for metric spaces and its applications for metrical task systems and related problems , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[11]  R. Ravi,et al.  On approximating planar metrics by tree metrics , 2001, Inf. Process. Lett..

[12]  Santosh S. Vempala,et al.  On Euclidean Embeddings and Bandwidth Minimization , 2001, RANDOM-APPROX.

[13]  Aravind Srinivasan,et al.  New approaches to covering and packing problems , 2001, SODA '01.

[14]  Aravind Srinivasan,et al.  Approximation algorithms for the covering Steiner problem , 2002, Random Struct. Algorithms.

[15]  Suchitra Amalapurapu,et al.  Approximation Algorithms for Directed Steiner Problems , 2002 .

[16]  Samir Khuller,et al.  On directed Steiner trees , 2002, SODA '02.

[17]  Guy Kortsarz,et al.  An approximation algorithm for the group Steiner problem , 2002, SODA '02.

[18]  Robert Krauthgamer,et al.  Polylogarithmic inapproximability , 2003, STOC '03.

[19]  Aravind Srinivasan,et al.  On the Covering Steiner Problem , 2003, FSTTCS.

[20]  Yair Bartal,et al.  Multi-embedding and path approximation of metric spaces , 2003, SODA '03.

[21]  Satish Rao,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2003, STOC '03.

[22]  E. Halperin,et al.  Asymmetric k-center is log* n-hard to approximate , 2004, STOC '04.

[23]  Yair Bartal,et al.  Multiembedding of Metric Spaces , 2004, SIAM J. Comput..

[24]  Santosh S. Vempala,et al.  Flow metrics , 2004, Theor. Comput. Sci..

[25]  Kunal Talwar,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2004, J. Comput. Syst. Sci..

[26]  Sudipto Guha,et al.  Asymmetric k-center is log* n-hard to approximate , 2005, JACM.