Changes
[http://www.google.com/books?id=PuL2KC_KCj0C&lpg=PA145&ots=7vnYFfZ-gu&dq=W.%20Mader%2C%20A%20reduction%20method%20for%20edge-connectivity%20in%20graphs%2C%20Ann.%20Discrete%20Math.%203%20(1978)%2C%20145-164.&lr=&pg=PA145#v=onepage&q=&f=false Google Books link].</ref> that the conjecture holds if the degree of every node in ''V-T'' is even. In <ref>A. Frank, T. Király, M. Kriesell, ''On decomposing a hypergraph into k connected sub-hypergraphs'', Discrete Applied Mathematics 131 (2003), 373-383, [http://dx.doi.org/10.1016/S0166-218X(02)00463-8 DOI link]. Preliminary version: [http://www.cs.elte.hu/egres/tr/egres-01-02.pdf EGRES Technical Report no. 2001-02].</ref> it was proved that if ''G'' is ''3k''-edge-connected and ''V-T'' is stable, then ''G'' contains ''k'' edge-disjoint Steiner trees for ''T''. Other bounds have been proved by Petingi and Rodriguez <ref>L. Petingi, J. Rodriguez, ''Bounds on the maximum number of edge-disjoint Steiner trees of a graph'', Congressus Numerantium 145 (2000) 43-52. [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.31.6991&rep=rep1&type=pdf Citeseer link].</ref>, and Jain et al.<ref>K. Jain, M. Mahdian, M.R. Salavatipour, ''Packing Steiner Trees'', SODA 2003. http://www.cs.ualberta.ca/~mreza/research/steiner-soda.ps</ref>.
A breakthrough result by Lau <refname="Lau">L. C. Lau, ''An Approximate Max-Steiner-Tree-Packing Min-Steiner-Cut Theorem'', Combinatorica 27 (2007), 71-90. [http://dx.doi.org/10.1007/s00493-007-0044-3 DOI link], [http://www.cse.cuhk.edu.hk/~chi/papers/steiner.pdf Author link].</ref> was that if ''G'' is ''26k''-edge-connected it in ''T'' then it has ''k'' edge-disjoint Steiner trees. This gave the first constant factor approximation result for the edge-disjoint Steiner trees problem. Petingi and Talafha <ref>L. Petingi, M. Talafha, ''Packing the Steiner trees of a graph'', Networks 54 (2009), 90-94, [http://dx.doi.org/10.1002/net.2029 DOI link].</ref> showed that the conjecture is true if the edge set of ''G'' can be partitioned into Steiner trees, and for general graphs they obtained that <math>(2k+1+|V-T|)</math>-edge-connectivity is sufficient. ===Hardness results=== Cheriyan and Salavatipour <ref>J. Cheriyan, M.R. Salavatipour, ''Hardness and Approximation Results for Packing Steiner Trees'', Algorithmica 45 (2006), 21-43. [http://dx.doi.org/10.1007/s00453-005-1188-4 DOI link], manuscript. [http://www.math.uwaterloo.ca/~jcheriya/PS_files/packsteiner-esa.ps Author link].</ref> proved that the problem of finding maximum number of edge-disjoint (or vertex-disjoint) Steiner trees is APX-hard even if there are only 4 terminals. They also showed that the directed version is hard to approximate within factor of <math>O(n^{\frac{1}{3}-\epsilon})</math> for any <math>\epsilon>0</math>, and gave an algorithm with approximation guarantee of <math>O(n^{\frac{1}{2}+\epsilon})</math>. Kaski <ref>P. Kaski, ''Packing Steiner trees with identical terminal sets'', Information Processing Letters 91 (2004), 1-5, [http://dx.doi.org/10.1016/j.ipl.2004.03.006 DOI link].</ref> proved that it is NP-complete to decide whether a graph contains 2 edge-disjoint Steiner trees for ''T''. ===Recent developments=== Recently Wu and West <ref>D.B. West, H. Wu, ''Packing of Steiner trees and S-connectors in graphs'', [http://www.math.uiuc.edu/~west/pubs/strees.pdf Author link].</ref> improved Lau's result <ref name="Lau"/> by showing that there are ''k'' edge-disjoint Steiner trees if ''G'' is <math>6.5 k</math>-edge-connected in ''T''.
==References==
