Past exam of the computer science course of the University of Oxford / 2018 / Algorithms / 2 / c / Solution

Root the tree arbitrarily. For each vertex $v$, let $D_1(v)$ be the minimum vertex cover size in its descendant subtree, conditional on selecting $v$, and $D_0(v)$ the corresponding minimum conditional on omitting $v$. Process the vertices in postorder traversal. For every child $u$ of $v$,At a leaf of a rooted tree, the empty sums give $D_1=1,D_0=0$. Return $\min \{D_0(r),D_1(r)\}$ at the root $r$.

For algorithm correctness, if $v$ is selected, each child subtree may independently use its cheaper state, since the connecting edge is already covered by $v$. If $v$ is omitted, each child must be selected to cover its connecting edge. Distinct child subtrees have no edges between them because the input is a tree. These observations prove the recurrences by structural induction. Each edge contributes constant work once, giving $O(|V|)$ time complexity and $O(|V|)$ space with adjacency lists. The single-vertex tree correctly has optimum zero. This is minimum vertex cover of a tree by dynamic programming.