Problem: Maximum Independent Set
INSTANCE: Graph $G=\left(V,E\right)$.
SOLUTION: An independent set of vertices, i.e. a subset $V' \subseteq V$ such that no two vertices in $V'$ are joined by an edge in $E$.
MEASURE: Cardinality of the independent set, i.e., $\vert V'\vert$.
$\max \sum x_v$
s.t.
$x_u + x_v \le 1 \quad \forall (u,v) \in E$
$x_v \in \{0,1\}$
(intuitively: choose as many nodes as you can, but for each edge in the graph, you can't choose both of its nodes).