Better approximation for special case of 3-hitting set I have a question based on 3-Hitting Set problem. In this problem, we are given a universal set U of size n and a set of subsets S such that $\forall $ s $\in$ S |s|&lt;=3.
FOr this problem, Integer programming formulation is given by 
$x_{a}+x_{b}+x_{c} >=1 $ (a,b,c) $\in$ S 

$\forall x_{i} \in $(1,0) where $i=1,2 \ldots n $

minimize $\sum x_{i} $

where each variable $x_i$ corresponds to element $i\in$ U.

A Linear programming relaxation would be 
0&lt;=$x_{i}$=&lt;1 instead of $x_{i} \in$ {1,0}  

After performing linear  programming we know that
In Every s $\in$S x_{a}+x_{b}+x_{c} >=1  at least one of  $x_{i}>=\frac{1}{3}$ 
so $x_{i}>=\frac{1}{3}$   is made as 1 and the corresponding elements are selected in the solution. There may exist a set s such that  $x_{a}+x_{b}+x_{c} >=1$ so that $x_{a},x_{b},x_{c}>=\frac{1}{3}$ and we will be picking all three elements. This way, we will get a three approximation after rounding $x_{a},x_{b},x_{c}$ to 1.

My question is 

I look at a  special case of 3 -hitting set H' in which size of each set s is exactly 3 and optimal solution for linear programming problem is $ \forall x_i  =\frac{1}{3} $. Now, is it possible to improve the approximation factor to less than 3
What you've defined is the vertex cover problem for 3-uniform hypergraphs. Khot and Regev showed that if the unique games conjecture is true, then it is NP-hard to get an approximation of $3-\epsilon$ for this problem.