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The minimum spanning tree problem can be solved in polynomial time via Kruskal's or Prim's algorithm. However, every integer program I have seen that corresponds to the MST problem require a constraint for every subset of vertices, so the number of constraints would be exponential.

That makes me wonder, is it is possible to write down the constraints for a IP corresponding to MST such that the number of constraints is polynomial in the number of vertices? Furthermore, if it is possible, does it hold that every problem in P can be written as a LP with at most polynomial number of constraints?

Thank you very much!

stensootla
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Yes, it's possible.

It's trivially possible that every problem in P can be reduced to LP in polynomial time, for uninteresting reasons: given an instance of the decision problem, the reduction solves the decision problem (in polynomial time), then either outputs a feasible LP instance or an infeasible LP instance (where these instances are hardcoded). I realize that's pretty uninteresting and probably not what you wanted.

D.W.
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There are IP formulations for MST with polynomially many constraints, see, e.g., "Optimal trees" by Magnati and Wolsey. For example, you can use the max-flow/min-cut theorem to find an equivalent formulation to the cut-based one that you mentioned: Replace each edge by two anti parallel arcs. Select one vertex as the root. Require that for each other vertex there is a unit flow from the root to this vertex. Very surprisingly, there are problems in P that you provably cannot formulate as an LP with polynomially many constraints. I would recommend the book "Integer programming" by Conforti et. al. for more Details.

Dan
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