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I am reading Slavik's paper A Tight Analysis of Greedy Set Cover. Specifically, Slavik considers the unweighted version of set cover where the ground set is ${U = \{1,\ldots,m\}}$ for arbitrarily large $m$. There is a collection of subsets ${S = \{S_1,\ldots,S_n\}}$ that is a cover of $U$. The goal is to find the smallest subcover ${S^* \subset S}$ that is a cover of $U$.

Let ${\textrm{Opt}:= \vert S^* \vert}$ be the size of the optimal solution and let $G$ be the size of the solution found by the traditional greedy algorithm that selects the subset that covers the most remaining elements. Slavik proves a bound that (see Eqs. (1) and (2))

$$ \ln m - \ln\ln m - 0.31 < \frac{G}{\textrm{Opt}} < \ln m - \ln\ln m + 0.78$$

What confuses me is how the approximation ratio can be lower bounded by a function of $m$. It seems to me that for arbitrary $m$, it would be possible for ${\frac{G}{\textrm{Opt}} = 1}$ (e.g., let $S_1 \in S$ be set ${\{1,\ldots,m-1\}}$ and let ${S_2 \in S}$ be set ${\{m\}}$ in which case it seems ${G = \textrm{Opt} = 2}$).

Clearly, I am missing something fundamental. Am I misunderstanding the lower bound on the approximation ratio entirely?

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Yes. You should read the lower bound as: there exists an instance with $m$ items such that $\frac{G}{\text{Opt}} \ge \log m - \log \log m - 0.31$.

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    $\begingroup$ Alternatively, Slavik is interested in the worst possible approximation ratio, as a function of $m$. $\endgroup$ Commented Jul 16, 2022 at 20:42

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