I have an LP with a (non-fixed) number of logical constraints in the form of $X_1 \rightarrow X_2$ (where $X_1$ and $X_2$ are linear functions inequalities of the $n$ input variables). To express these logical operations I introduce binary variables (as in here) $\gamma_1$, $\gamma_2$ so I end up with a Mixed Integer Linear Program that looks something like this:
$$ \begin{equation*} \begin{aligned} & \underset{x, \gamma}{\text{minimize}} & & c^T \mathbf{x} \\ & \text{subject to} & & X^i_{1} \leq \gamma^i_1, \; i \in [n], \\ &&& X^i_{2} \leq \gamma^i_2, \; i \in [n], \\ &&& \gamma^i_1 \leq \gamma^i_2, \; i \in [n], \\ &&& \gamma^i_1, \gamma^i_2 \in \left\{ 0,1\right\}, \; i \in [n] \\ &&& \mathbf{x} \succeq 0 \end{aligned} \end{equation*} $$
I know these types of problems can be solved well in practice using a wide range of heuristics, but I am interested in a provably efficient algorithm for approximating this problem.
I don't see why the LP relaxation of this MILP (obtained by only asking that the $\gamma$'s be in $[0,1]$) gives me anything: unlike classic examples like Set Cover, is it clear how to obtain a feasible solution (integral) to the original problem from a solution to the LP relaxation? A rounding approach seems useless here, because I can do whatever I want to the values of $\gamma$, but that doesn't actually effect the solution $\mathbf{x}$. If I can't "translate" a solution to the LP to an integral solution, how can I say anything about the integrality gap?
My questions:
- Am I correct in that for this particular type of MILP (resulting from logical constraints), the LP relaxation is useless?
- Is there any other approach which I might be able to use to say something about the ability to approximate my problem?
I have very little background in approximation algorithms so I would really appreciate any pointers. I hope the question is clear and made sense. Thanks!
