Cs 598csc: Approximation Algorithms
نویسنده
چکیده
Let G = (V,E) be an undirected graph with arc weights w : V → R+. Define xv for each vertex v as follows: xv = 1, if v is in the vertex cover; xv = 0, if v is not chosen. Our goal is to find min ( ∑ v∈V wvxv), such that xu + xv ≥ 1, ∀e = (u, v) ∈ E, xv ∈ {0, 1}. However, we can’t solve Integer Linear Programming (ILP) problems in polynomial time. So we have to use Linear Programming (LP) to approximate the optimal solution, OPT(I), produced by ILP. First, we can relax the constraint xv ∈ {0, 1} to xv ∈ [0, 1]. It can be further simplified to xv ≥ 0, ∀v ∈ V . Thus, a Linear Programming formulation for Vertex Cover is:
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