Is this a proof that SET COVER is not an NP-hard problem? In this paper, Karpinski and Zelikovsky introduce the SET COVER and the $\epsilon$-DENSE SET COVER problems as follows:


  Set Cover Problem. Let $X = \{x_1, \ldots, x_k\}$ be a finite set and $P = \{p_1, \ldots, p_m\} \subseteq 2^X$ be a family of its
  subsets. Find minimum size sub-family $M$ of $P$ such that
  $X \subseteq \bigcup \{p \mid p \in M\}$.
  
  An instance of the set cover problem is $\epsilon$-dense if there is
  $\epsilon &gt; 0$ such that any element of $X$ belongs to at least
  $\epsilon \, m$ sets from $P$. We show that the dense set cover
  problem can be approximated with the performance ratio $c \log k$ for
  any $c &gt; 0$ though it is unlikely to be NP-hard.


In page 3, Lemma 2.1 implies—and Theorem 2.2 proves—that $\epsilon$-DENSE SET COVER is not NP-hard. But, since SET COVER is $\frac{1}{m}$-DENSE SET COVER (any element of $X$ belongs to at least 1 of the $m$ sets), would this mean that SET COVER is not NP-hard?
Weighting of elements from set $P$ of SET COVER problem can be obtained by solving a system of linear inequalities for a given $\varepsilon$ that involves $\{0,1\}$-matrix $I=[x_i\in P_j]:$ $Iw\geq\varepsilon\sum\limits_{i=1}^mw_i.$ If it is not feasible then $\varepsilon$-DENSE SET COVER is void. Found weighting can be easily removed by repeating elements of $P$ with respect to the weighting but the resulting problem will be the same. Feasible SET COVER could be reduced to dense one for $\varepsilon=\frac{1}{m}$ but generally aforementioned reduction could lead to void dense problem for constant $\varepsilon.$

In the geometric case when VC-dimension is finite the dense problem with constant $\varepsilon$ could be solved in polynomial time due to result of Haussler and Welzl "epsilon-nets and simplex range queries" claiming existence of feasible solution to $\varepsilon$-dense problem of length $$\frac{8d}{\varepsilon}\log\frac{d}{\varepsilon}.$$
Having upper bound on the problem optimum we could solve te problem by brute-force search. Here $d$ denotes VC-dimension.