Solve Steiner Forest on T, let E∗T ⊆ ET be the solution on T.

There are many applications of the tree embedding theorem in approximation and online algorithms. It allows one to reduce certain class of problems on graphs to problems on trees and it is much easier to design algorithms on trees. The ratio one obtains using this approach might not be optimal but gives one a quick upper bound on the approximation ratio. For some problems this is still the best approach known. Let’s recall Steiner Forest problem. Given edge weighted graph G = (V,E) and pairs of nodes {s1t1, s2t2,..., sktk}, the object is to find minimum cost subset E ′ ⊆ E such that each siti is connected in G[E ′]. In pervious lecture, we have obtained a 2(1 − 1 k) approximation for Steiner Forest using LP based methods, e.g. primal-dual method. However, we can obtain a trivial algorithm by using tree embedding technique. Here is the algorithm: Steiner Forest(G(V,E)): Pick a tree T = (V,ET) randomly from tree embedding of G