Past exam of the computer science course of the University of Oxford / 2018 / Algorithms / 3 / b / Solution

Let $Z$ be the set of isolated vertices of the finite simple graph $G=(V,E)$. Introduce a nonnegative integer $x_v$ for every vertex. Use the integer programEvery nonisolated vertex belongs to an edge, whose inequality and the other endpoint's nonnegativity already imply $x_v\le 1$. The explicit bounds on $Z$ therefore make all variables binary. For an integer feasible point, $S=\{v:x_v=1\}$ is an independent set, and its cardinality is the objective. Conversely the indicator vector of any independent set is feasible. Thus the optimal value is the maximum independent set size. Including the isolated-vertex bounds is essential for an arbitrary graph; without them the formulation would be unbounded whenever an isolated vertex is present.