By Trevisan L.

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This publication constitutes the joint refereed lawsuits of the 14th foreign Workshop on Approximation Algorithms for Combinatorial Optimization difficulties, APPROX 2011, and the fifteenth foreign Workshop on Randomization and Computation, RANDOM 2011, held in Princeton, New Jersey, united states, in August 2011.

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**Sample text**

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.