Does real linear programming produce bipartite perfect matching using maxflow reduction?

cs.stackexchange https://cs.stackexchange.com/questions/109743

質問

Given a bipartite graph the standard reduction to max flow is with the construction similar to following diagram:

enter image description here

We can formulate max flow as an linear programming problem with integer variables in latter.

  1. If we do not use integer variables does solving for maxflow in linear programming formulation with only real variables still produce valid perfect matching of given graph?

  2. Is there a formal proof of this?

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