i:ei Sj in .NET Add Data Matrix barcode in .NET i:ei Sj

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i:ei Sj generate, create data matrix 2d barcode none for .net projects QR Code Safty Let be the smallest di DataMatrix for .NET erence between the right-hand side and left-hand side of all constraints ( ) involving ek ; that is, = minj:ei Sj wj i:ei Sj yi . By inequality (1.

4), we know that > 0.. Consider now a new d ual solution y in which yk = yk + and every other component of y is the same as in y . Then y is a dual feasible solution since for each j such that ek Sj , yi = yi + wj , i:ei Sj i:ei Sj. by the de nition of . For each j such that ek Sj , / yi wj , yi = i:ei Sj i:ei Sj as before. Furth datamatrix 2d barcode for .NET ermore, n yi > n yi , which contradicts the optimality of y .

Thus it i=1 i=1 must be the case that all elements are covered and I is a set cover. Theorem 1.8: The dual rounding algorithm described above is an f -approximation algorithm for the set cover problem.

Proof. The central idea is the following charging argument: when we choose a set Sj to be in the cover we pay for it by charging yi to each of its elements; each element is charged at most once for each set that contains it (and hence at most f times), and so the total cost is at most f m yi , or f times the dual objective function. i=1 More formally, since j I only if wj = i:ei Sj yi , we have that the cost of the set cover I is wj = yi.

j I j I i:ei Sj n i=1 n fi yi { } . j I : ei Sj yi i=1 n i=1 yi f OPT . Electronic w data matrix barcodes for .NET eb edition.

Copyright 2010 by David P. Williamson and David B. Shmoys.

To be published by Cambridge University Press. Constructing a dual solution: the primal-dual method The second equality foll visual .net data matrix barcodes ows from the fact that when we interchange the order of summation, the coe cient of yi is, of course, equal to the number of times that this term occurs overall. The nal inequality follows from the weak duality property discussed previously.

In fact, it is possible to show that this algorithm can do no better than the algorithm of the previous section; to be precise, we can show that if I indexes the solution returned by the primal rounding algorithm of the previous section, then I I . This follows from a property of optimal linear programming solutions called complementary slackness. We showed earlier the following string of inequalities for any feasible solution x to the set cover linear programming relaxation, and any feasible solution y to the dual linear program:.

n i=1 yi n i=1 j:ei Sj xj = m j=1 i:ei Sj yi m j=1 xj wj . Furthermore, we claimed that strong duality implies that for optimal solutions x and y , n m i=1 yi = j=1 wj xj . Thus for any optimal solutions x and y the two inequalities in the chain inequalities above must in fact be equalities. This can only happen if whenever yi > 0 of then j:ei Sj x = 1, and whenever x > 0, then i:ei Sj yi = wj .

That is, whenever a linear j j programming variable (primal or dual) is nonzero, the corresponding constraint in the dual or primal is tight. These conditions are known as the complementary slackness conditions. Thus if x and y are optimal solutions, the complementary slackness conditions must hold.

The converse is also true: if x and y are feasible primal and dual solutions respectively, then if the complementary slackness conditions hold, the values of the two objective functions are equal and therefore the solutions must be optimal. In the case of the set cover program, if x > 0 for any primal optimal solution x , then the j corresponding dual inequality for Sj must be tight for any dual optimal solution y . Recall that in the algorithm of the previous section, we put j I when x 1/f .

Thus j I implies that j j I , so that I I..
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