Minimum cost strongly connected digraph

algorithmgraphnp-hardtraveling-salesman

I have a digraph which is strongly connected (i.e. there is a path from i to j and j to i for each pair of nodes (i, j) in the graph G). I wish to find a strongly connected graph out of this graph such that the sum of all edges is the least.

To put it differently, I need to get rid of edges in such a way that after removing them, the graph will still be strongly connected and of least cost for the sum of edges.

I think it's an NP hard problem. I'm looking for an optimal solution, not approximation, for a small set of data like 20 nodes.

Edit

A more general description: Given a grap G(V,E) find a graph G'(V,E') such that if there exists a path from v1 to v2 in G than there also exists a path between v1 and v2 in G' and sum of each ei in E' is the least possible. so its similar to finding a minimum equivalent graph, only here we want to minimize the sum of edge weights rather than sum of edges.

Edit:

My approach so far:
I thought of solving it using TSP with multiple visits, but it is not correct. My goal here is to cover each city but using a minimum cost path. So, it's more like the cover set problem, I guess, but I'm not exactly sure. I'm required to cover each and every city using paths whose total cost is minimum, so visiting already visited paths multiple times does not add to the cost.

Best Solution

Your problem is known as minimum spanning strong sub(di)graph (MSSS) or, more generally, minimum cost spanning sub(di)graph and is NP-hard indeed. See also another book: page 501 and page 480.

I'd start with removing all edges that don't satisfy the triangle inequality - you can remove edge a -> c if going a -> b -> c is cheaper. This reminds me of TSP, but don't know if that leads anywhere.

My previous answer was to use the Chu-Liu/Edmonds algorithm which solves Arborescence problem; as Kazoom and ShreevatsaR pointed out, this doesn't help.

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