Combinatorial optimization: Exact and approximate algorithms by Trevisan L.

By Trevisan L.

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4) This is the kind of expression for which the following trick is useful: 1 − x is approximately e−x for small values of x. More precisely, using the Taylor expansion around 0 of ex , we can see that, for 0 ≤ x ≤ 1 e−x = 1 − x + x2 x3 − + ··· 2 3!

5. LINEAR-TIME 2-APPROXIMATION OF WEIGHTED VERTEX COVER 53 ∗ y(u,v) := 1 ∗ xu := 1 ∗ xv := 1 • S := {v : xv = 1} • return S, y Our goal is to modify the above algorithm so that it can deal with vertex weights, while maintaining the property that it finds an integral feasible x and a dual feasible y such that v∈V c(v)xv ≤ 2 · (u,v)∈V yu,v . 5). We will get simpler formulas if we think in terms of a new set of variables pv , which represent how much we are willing to “pay” in order to put v in the vertex cover; at the end, if pv = cv then the vertex v is selected, and xv = 1, and if pv < cv then we are not going to use v in the vertex cover.

So it must be the case that opt ≥ n−i+1 = (n − i + 1) · cost(xi ) ci from which we get n apx ≤ opt · i=1 1 n−i+1 The quantity n i=1 1 = n−i+1 n i=1 1 i is known to be at most ln n + O(1), and so we have apx ≤ (ln n + O(1)) · opt It is easy to prove the weaker bound n 1 i=1 n ≤ log2 n + 1 , which suffices to prove that our algorithm is O(log n)-approximate: just divide the sum into terms of the form that is 1+ 1 1 + 2 3 + 1 1 1 1 + + + 4 5 6 7 2k+1 −1 1 i, i=2k + ··· and notice that each term is at most 1 (because each term is itself the sum of 2k terms, each ≤ 2−k ) and that the whole sum contains at most log2 n + 1 such terms.

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