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I am reading the book "Approximation Algorithm" by Vijay V Vazirani http://athena.nitc.ac.in/~kmurali/Courses/CombAlg2014/vazirani.pdf

And in proof of Lemma 2.3, the author said "In the iteration in which $e_k$ was covered, $\overline{C}$ contained at least $n-k+1$ elements." without any claims.

But this is not so trivial for me. Can anyone give me some hint or explain?

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This is based on the fact that you update the sets in the following in each iteration in the following manner: Let $\hat{S}_i$ be the set with the minimum cost-effectiveness. Then update for all sets which indices are not yet added to the solution index set $\hat{S}_{j}= \hat{S}_{j} - \hat{S}_{i}$ $\forall j \neq i$. Therefore no set will contain the elements which are already added and hence in each iteration at least one new element is covered by the solution. Then you have trivially $\frac{1}{n-k+1}$ elements left at the start of the $k$-th iteration.

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