The asymmetric integer program reuses the same logic on a different feasible region. Its LP relaxation and integer optimum do not match. This lesson shows the method generalizing without changing the exact arithmetic.

highlighted = computed this step

LP is not IP

The LP relaxation is (6/5, 8/5) with z=22/5. Why: the relaxed point is a bound, not an integer solution.

zLP=22/5z_{LP}=22/5
IP-A treeA second branch-and-bound tree proves the asymmetric integer optimum.IP-A treex<=1y<=1y>=2x>=2rootLP (6/5, 8/5)z=22/5branchedx<=1LP (1, 5/3)z=13/3branchedy<=1LP (1, 1)z=3integer candidatey>=2LP (0, 2)z=4incumbent optimalx>=2LP (2, 0)z=2integer candidate

Second integer certificate

The integer optimum is (0, 2) with z=4. Why: the same branching logic generalizes to a different IP.

zIP=4z_{IP}=4
IP-A treeA second branch-and-bound tree proves the asymmetric integer optimum.IP-A treex<=1y<=1y>=2x>=2rootLP (6/5, 8/5)z=22/5branchedx<=1LP (1, 5/3)z=13/3branchedy<=1LP (1, 1)z=3integer candidatey>=2LP (0, 2)z=4incumbent optimalx>=2LP (2, 0)z=2integer candidate

Diagram note

The second tree is recomputed from the asymmetric IP, not copied from IP-B. Pixel positions are rounded for layout; every number shown is exact.

integer optimum differs from LP relaxation\text{integer optimum differs from LP relaxation}
IP-A treeA second branch-and-bound tree proves the asymmetric integer optimum.IP-A treex<=1y<=1y>=2x>=2rootLP (6/5, 8/5)z=22/5branchedx<=1LP (1, 5/3)z=13/3branchedy<=1LP (1, 1)z=3integer candidatey>=2LP (0, 2)z=4incumbent optimalx>=2LP (2, 0)z=2integer candidate