Minimum True Monotone 3SAT I am interested in a SAT variation where the CNF formula is monotone (no variables are negated). Such a formula is obviously satisfiable.

But say the number of true variables is a measure of how good our solution is.  So we have the following problem:

MINIMUM TRUE MONOTONE 3SAT

INSTANCE: Set U of variables, collection C of disjunctive clauses of 3 literals, where a literal is a variable (not negated).
SOLUTION: A truth assignment for U that satisfies C.
MEASURE:  The number of variable that are true.

Could someone give me some helpful remarks on this problem?
This problem is the same as the Vertex Cover problem for $3$-uniform hypergraphs: 
given a collection $H$ of subsets of $V$ of size $3$ each, find a minimal subset $U\subseteq V$ that intersects each set in $H$.

It is therefore NP-hard, but fixed parameter tractable. 
It is also NP-hard to approximate to within a factor of $2-\epsilon$ for every $\epsilon&gt;0$. This was shown in the following paper:

Irit Dinur, Venkatesan Guruswami, Subhash Khot and Oded Regev.
A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover,
SIAM Journal on Computing, 34(5):1129–1146, 2005.