Minimum spanning tree formulation as integer program
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.