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I've encountered a description of a well-known, hard to compute problem. Let us consider $n$ sites that need to be connected in the shortest possible way. If I am only allowed to connect the sites I have a traveling salesman problem. Now, I am allowed to add nodes to the map and connect the sites to those nodes (or even connect two nodes). What it the name of this problem?

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It's hard to understand what you're asking here. You said $n$ sites and that they have to be connected in the "shortest" way; are you given pairwise distances for all sites? When you add a new site, what are the distances from it to all other sites? –  Fixee Sep 20 '11 at 6:48

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Assuming the sites are given as points in the plane, this is the Steiner tree problem.

By the way, if you are trying to do the same thing without introducing additional nodes, as in the initial part of your question, then this is the problem of computing a minimum spanning tree, not the travelling salesman problem, and can be solved in polynomial time.

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