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Can you fix integral LP variables for a non-integral polytope without affecting the existence of integral optima?

Suppose I have a linear program $LP1=\{\mbox{Maximize }c^\top x \mid x\in \mathcal{P}\}$ for some polytope $\mathcal{P}\subseteq [0,1]^n$, which is known to have fractional extreme points. Suppose ...
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