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## CS787: Advanced Algorithms Lecture 12: LP Duality and the Primal-Dual method

### Abstract

In the previous few lectures we have seen examples of LP-rounding, a method for obtaining approximation algorithms that involves solving a linear programming relaxation of the problem at hand and rounding the solution. Previously we also discussed the basic theory of LP-duality. Today we will apply this theory to obtain a second LP-based technique for obtaining approximation algorithms — the primal-dual method. This technique has the advantage that it circumvents the need to actually solve an LP relaxation, leading to efficient algorithms that are purely combinatorial. We will apply this technique to the Vertex Cover and Steiner Forest problems. The primal-dual method in the context of approximation algorithms was first used by Goemans and Williamson [1]. 12.1 Linear Programming Duality Consider the general linear program minimize subject to cT x Ax ≥ b x ≥ 0 where x = (x1,..., xn) T is vector of variables, A = (Aij) is an m × n matrix, and c = (c1,..., cn) T and b = (b1,..., bm) T. Recall that the dual of this linear program is given by maximize subject to bT y AT y ≤ c y ≥ 0 where y = (y1,..., ym) T is a vector of variables. The variables in y are in one-to-one correspondence with the constraints of the primal LP, and the variables in x are in one-to-one correspondence with the constraints of the dual LP. In a previous lecture we showed that the dual of the dual of an LP is again the original LP. We also proved the following result, stating that the objective value of every feasible solution to the dual lower bounds the objective value of every feasible solution to the primal. We reiterate the proof since it will come in handy later. Theorem 12.1.1 (Weak Duality Theorem) If x and y are feasible solutions to the primal and dual respectively, then c T x ≥ b T y. Proof: We have c T x = n∑ i=1 cixi ≥ n ∑ ( ∑m i=1 j=

Year: 2011
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