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I know the constraints matrix $A$ of a linear program $$ \min c^Tx \text{ subject to } b\le Ax $$ is totally unimodular. So, the program has integral solutions for integral vector $b$.

Is this is also the case for the following problem: $$ \min c^Tx \text{ subject to } b_1\le Ax\le b_2 $$ where $b_1$ and $b_2$ are integral vectors and $A$ is totally unimodular. Does it have integral solutions, too?

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The problem can be converted into standard form using slack variables $s_1$ and $s_2$. Then the two-sided problem is equivalent to $$ \min c^Tx $$ subject to $$ \pmatrix{ A & -I &0\\ A& 0&I}\pmatrix{x\\s_1\\s_2} = \pmatrix{b_1\\b_2} $$ and $$ x\ge0, \ s_1\ge0, \ s_2 \ge0. $$ The matrix appearing above is totally unimodular:

  • If $A$ is totally unimodular, then is $\pmatrix{-A\\A}$.
  • If $B$ is totally unimodular, so is $\pmatrix{B & I}$.
  • If $\pmatrix{C\\D}$ is totally unimodular, so is $\pmatrix{-C\\D}$.

These facts together prove that the system matrix above is totally unimodular as well.

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