CS 598 CSC : Approximation
نویسندگان
چکیده
In the last lecture a primal-dual analysis was used to show a 2-approximation for theVertex Cover problem. Here we extend these ideas to show a 2-approximation for the Steiner Forest problem using a primal-dual analysis. In the Steiner Forest problem there is a graph G = (V,E) where each edge e ∈ E has a cost ce ∈ R. We are given k pairs of terminals s1, t1, s2, t2, . . . sk, tk ∈ V . The goal is to find the minimum cost set of edges F ∗ such that in the graph G[F ∗] the vertices si and ti are in the same connected component for 1 ≤ i ≤ k. Here G[F ∗] denotes the induced subgraph G on the set of edges F ∗. Notice that G[F ∗] can contain multiple connected components. We now want to transform this problem into a IP. First we set up some notation. In the IP we will have a variable xe for each edge e ∈ E such that xe is 1 if and only if e is part of the solution. Let the set S be the collection of all sets S ⊂ V such that S disconnects si and ti for some 1 ≤ i ≤ k. For a set S ⊂ V let δ(S) denote the set of edges crossing the cut (S, V \ S). The IP can be written as the following.
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