Weighted k‐cardinality trees: Complexity and polyhedral structure

We consider the k‐CARD TREE problem, i.e., the problem of finding in a given undirected graph G a subtree with k edges, having minimum weight. Applications of this problem arise in oil‐field leasing and facility layout. Although the general problem is shown to be strongly NP hard, it can be solved in polynomial time if G is itself a tree. We give an integer programming formulation of k‐CARD TREE and an efficient exact separation routine for a set of generalized subtour elimination constraints. The polyhedral structure of the convex hull of the integer solutions is studied.