Question

I have Graph with N nodes and edges with cost. (graph may be Complete but also can contain zero edges).

I want to find K trees in the graph (K < N) to ensure every node is visited and cost is the lowest possible.

Any recommendations what the best approach could be?

I tried to modify the problem to finding just single minimal spanning tree, but didn’t succeeded.
Thank you for any hint!

EDIT

little detail, which can be significant. To cost is not related to crossing the edge. The cost is the price to BUILD such edge. Once edge is built, you can traverse it forward and backwards with no cost. The problem is not to “ride along all nodes”, the problem is about “creating a net among all nodes”. I am sorry for previous explanation

The story
Here is the story i have heard and trying to solve.

There is a city, without connection to electricity. Electrical company is able to connect just K houses with electricity. The other houses can be connected by dropping cables from already connected houses. But dropping this cable cost something. The goal is to choose which K houses will be connected directly to power plant and which houses will be connected with separate cables to ensure minimal cable cost and all houses coverage 🙂

Answer 1

I just came up with the easy solution as follows:

N – node count

C – direct connections to the grid

E – available edges

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1, Sort all edges by cost

2, Repeat (N-C) times:

1. Take the cheapest edge available
2. Check if adding this edge will not caused circles in already added edge
3. If not, add this edge

3, That is all… You will end up with C disjoint sets of edges, connect every set to the grid