Setup

An integer linear$-$programming problem is a linear$-$programming problem with the additional constraint that the variables $|$$$x|$$ must take on integer values. Turns out there is no known polynomial$-$time algorithm for this problem.

Part A

Show that weak duality $($Lemma $29\mathrm{.}1$ from the reading$)$ holds for an integer linear program.

Part B

Show that duality $($Theorem $29\mathrm{.}4$ from the reading$)$ does not always hold for an integer linear program.

Part C

Given a primal linear program in standard form, let $\frac{\frac{?}{?}}{\text{\$}}2$$ be the optimal objective value for the primal linear program, $|$$$D|$$ be the optimal objective value for its dual, $|$$IP$|$$ be the optimal objective value for the integer version of the primal $($that is$,$ the primal with the added constraint that the variables take on integer values$),$ and $|$$$ID|$$ be the optimal objective value for the integer version of the dual. Assuming that both the primal integer program and the dual integer program are feasible and bounded, show that $|$$$IP|P=DI\frac{D}{\frac{?}{?}}$$$.$