How to find weighted minimum vertex cover of graph

Question

So, I have a graph of vertices which have certain weights and edges. I'm trying to find the minimum weighted vertex cover. For example if I have a vertex cover of size 10 but each node has a weight of 10, then the weight of the total cover is 100. But if I have a vertex cover of size 99 with each node of weight 1, then I would pick this cover over the previous one.

This is NP-Complete I believe, so there's no efficient algorithm, but I think even an exhaustive search would work for me because the number of nodes will be relatively small. The only way I can think to do this then would be to generate the power set of the set [1 ... n] (where each integer corresponds to a node on the graph), then test every individual set to see if it is 1) a valid vertex cover, and 2) keep track of the vertex cover of lowest weight.

But this seems horribly inefficient. Is this the best way to go about it?

Answer 1

Minimum weight vertex cover is NP-Complete so you couldn't expect better than exhaustive search in general, but you could use backtracking to find a minimum weight vertex cover, something like this :

MinCover(Graph G, List<Vertex> selectedVertices, int min)
{
   var coveredAll = covered(G,selectedVertices);
   if ( coveredAll && weight(selectedVertices) < min)
   {
       cover = selectedVertices.ToList();
       min = weight(cover);
   }
   else if (!coveredAll && weight(selectedVertices) < min)
   {
      select another unvisited vertex and add it to selectedVertices
      call MinCover
      remove the previously selected vertex from the list
   }

   return;

}