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Consider Maximum-Coverage problem, which means there are set $A$ and $n$ subsets of $A$, we can choose $k$ subsets to cover some elements, and Maximum-Coverage is the assignment that cover most elements in $A$. We know by LP relaxation there is $1-1/e$ approximation algorithm, however, the integrality gap for it is $1-1/e+\epsilon$, which means there is an instance $I$ such that $LP(I)=1$ and $Opt(I)<1-1/e+\epsilon$, how to choose the problem set $I$?

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  • $\begingroup$ What do you mean by $LP(I) = 1$? $\endgroup$
    – codeR
    Commented May 10 at 7:24
  • $\begingroup$ If the integrality gap is $1-\frac{1}{e} + \epsilon$, then by definition we have $LP(I) \ge \frac{OPT(I)}{1-\frac{1}{e} + \epsilon}$. $\endgroup$
    – codeR
    Commented May 10 at 7:27
  • $\begingroup$ What do you mean by "choose the problem set $I$"? Do you mean you want to find an instance (not problem set) where this bound is tight? $\endgroup$
    – codeR
    Commented May 10 at 7:29
  • $\begingroup$ Yes, I want to find a counterexample. $\endgroup$ Commented May 10 at 12:25
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    $\begingroup$ I would recommend you modify your question to better reflect your requirements. $\endgroup$
    – codeR
    Commented May 10 at 12:33

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