So my problem is the following:
I have an undirected (complete) weighted graph G=(V,E), and I would like to generate all the possible spanning trees with minimum number of leaves, i.e. with minimum number of vertices of degree 1. Let's call this kind of trees MIN_LEAF.
Possibly, I would like to directly generate, among all trees with minimum number of leaves, the one which has also the minimum total weight (please note that this is not necessarily a minimum spanning tree). Is the problem of deciding if a tree T is a MIN_LEAF for a given graph G NP-complete?
If so, I wonder if some kind of heuristic algorithm exists (greedy or local search) which can at least give an approximate solution for this problem.
Thanks in advance.
The first problem you described - finding a spanning tree with the fewest number of leaves possible - is NP-hard. You can see this by reducing the Hamiltonian path problem to this problem: notice that a Hamiltonian path is a spanning tree of a graph and only has two leaf nodes, and that any spanning tree of a graph with exactly two leaf nodes must be a Hamiltonian path. That means that the NP-hard problem of determining whether a Hamiltonian path exists in a graph can be solved by finding the minimum-leaf spanning tree of the graph: the path exists if and only if the minimum-leaf spanning tree has exactly two leaves. The second problem you've described contains that first problem as a special case and therefore is going to also be NP-hard.
A quick Google search turned up the paper "On finding spanning trees with few leaves", which seems like it might be a good starting point for approximation algorithms (they have a 2-approximation for arbitrary graphs) and further reading on the subject.