Design of Approximation Algorithms

188 The primal-dual methodα 1 + α 2 = 1, with α 1 , α 2 ≥ 0. Note that this implies thatα 1 = k − |S 2||S 1 | − |S 2 | and α 2 = |S 1| − k|S 1 | − |S 2 | .We can then get a dual solution (ṽ, ˜w) by letting ṽ = α 1 v 1 + α 2 v 2 and ˜w = α 1 w 1 + α 2 w 2 .Note that (ṽ, ˜w) is feasible for the dual linear program (7.9) with facility costs λ 2 since it isa convex combination of two feasible dual solutions. We can now prove the following lemma,which states that the convex combination of the costs of S 1 and S 2 must be close to the cost ofan optimal solution.Lemma 7.15:α 1 c(S 1 ) + α 2 c(S 2 ) ≤ (3 + ϵ) OPT k .Proof. We first observe that⎛⎞c(S 1 ) ≤ 3 ⎝ ∑ vj 1 − λ 1 |S 1 | ⎠j∈D⎛⎞= 3 ⎝ ∑ vj 1 − (λ 1 + λ 2 − λ 2 )|S 1 | ⎠j∈D⎛⎞= 3 ⎝ ∑ vj 1 − λ 2 |S 1 | ⎠ + (λ 2 − λ 1 )|S 1 |j∈D⎛⎞≤ 3 ⎝ ∑ vj 1 − λ 2 |S 1 | ⎠ + ϵ OPT k ,j∈Dwhere the last inequality follows from our bound on the difference λ 2 − λ 1 .Now if we take the convex combination of the inequality above and our bound on c(S 2 ), weobtain⎛⎞α 1 c(S 1 ) + α 2 c(S 2 ) ≤ 3α 1⎝ ∑ vj 1 − λ 2 |S 1 | ⎠ + α 1 ϵ OPT kj∈D⎛⎞+ 3α 2⎝ ∑ vj 2 − λ 2 |S 2 | ⎠ .j∈DRecalling that ṽ = α 1 v 1 + α 2 v 2 is a dual feasible solution for facility costs λ 2 , that α 1 |S 1 | +α 2 |S 2 | = k, and that α 1 ≤ 1, we have that⎛α 1 c(S 1 ) + α 2 c(S 2 ) ≤ 3 ⎝ ∑ ṽ j − λ 2 k⎠ + α 1 ϵ · OPT k ≤ (3 + ϵ) OPT k .j∈D⎞Electronic web edition. Copyright 2011 by David P. Williamson and David B. Shmoys.To be published by Cambridge University Press