Randomized Rounding of Solutions to Linear Programs
Of course, if you round, you have to verify that rounding preserves feasibility.

Let us for example consider the relaxed VERTEX-COVER LP formulation.
$$
\begin{array}{lll}
\text{min} &amp; \sum_{v\in V}c(v)x_v &amp; \\
\text{s.t.} &amp;x_u+x_v\ge1, &amp; \quad (u,v)\in E \\
&amp;x_v\ge 0. &amp; \quad v\in V
\end{array}
$$

It is known that the solution to this problem is half-integral, i.e., each variable is either $0$, $1$, or $1/2$. The rounding scheme works as follows, whenever your solution contains $x_v=1/2$ you round up and set $x_v=1$. The constraints of the ILP were
$$
\begin{array}{lll}
 &amp;x_u+x_v\ge1, &amp; \quad (u,v)\in E \\
&amp;x_v \in \{0,1\}. &amp; \quad v\in V
\end{array}
$$
Both constraints are fulfilled after the rounding. And you have a nice 2-approximation.