The Complexity and Algorithm for Minimum Expense Spanning Trees

The minimum spanning tree problem is a classical and well-known combinatorial optimization problem. There exist many efficient algorithms such as the Kruskal algorithm and Prim algorithm to solve it. But in a real network, the vertices as well as the edges may have weights, and there are many cases of the vertex weights according to the degrees of the vertices. In this paper, we consider the computational complexity of the minimum expense spanning tree problem, which is to find a spanning tree in a network with minimum total expenses. We show that this problem is NP-hard in some general situations. And we propose a polynomial time algorithm when computing all the weights of the vertices in a spanning tree.