Let's say i have a linear integer program IP and let LP the linear relaxation of IP. I want to maximize IP and let $z_{IP} > 0$ the optimal value of IP and $z_{LP} > 0$ the optimal value of the linear relaxation LP. So it's clear that
$z_{IP} \le z_{LP}$
holds. But i think i can find a $\alpha > 0$ that
$z_{LP} \le \alpha \cdot z_{IP}$
for any IP holds, is this right? First, i thought i consider the dual of IP and its relaxation and do the same or use the weak (strong) duality but i dont get any good results.
Any idea or hint would be very kind. :)