Past exam of the integer programming course of the University of Oxford / 2011 / 1 / b / Solution

Use the maximum-weight version of Kruskal's algorithm: sort the edges by decreasing weight, start with no edges, and add an edge exactly when it joins different current connected components of a graph. For the maximum over all forests, skip negative-weight edges; removing a negative edge always preserves a forest and increases its weight. Zero-weight edges are optional. The matroid greedy algorithm for the graphic matroid proves this procedure optimal; it maximizes over independent sets rather than requiring a full basis.

The positive edges are considered in the orderAdmit $12$ and $15$. Reject $25$, since it closes the triangle on $1,2,5$. Admit $23$. Reject $35$, since $3$ and $5$ are already joined by the path $3,2,1,5$. Skip both negative edges incident with vertex four. ThereforeVertex four is isolated. Requiring a spanning tree would force a negative edge and solve a different optimization problem. With a disjoint-set data structure, sorting dominates the running time, giving $O(|E|\log |E|)$.