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Linear programs with a totally unimodular system matrix are known to have an optimal integer point. They are therefore solvable via relaxing the integer constraints to intervals.

An other interesting phenomenon occurs in linear programming relaxations to quadratic pseudo-Boolean functions. If those relaxation have an optimal point $x$ for which $x_i=0$, then there exists an optimal solution to the original problem $x^* $ for which $x^*_i=0$. The analogue holds for $x_i=1$. That is, even if the solution is not integral (Boolean), the points which are still tell something about the solution to the original problem.

Is there a general property of integer (0/1) programs, similar to total unimodularity, which enables one to draw conclusions about partially optimal solutions like this?

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Here is an example, where this property is proved for a very special type of integer program:

G.L. Nemhauser and L.E. Trotter, Jr.: Vertex Packings: Structural Properties and Algorithms, Mathematical Programming, 1975.

I am also interested in other examples.

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