Bound Principle for Integer Linear Programming and Linear Programming

Question

Currently, I am learning approximation algorithms. When I learned Vertex Cover via LP, I encountered a principle called Bounding Principles. It like this:

(1) The maximum value for an ILP problem is always less than or equal to the maximum value for the LP relaxation:

MAX for ILP ≤ MAX for LP relaxation

(2) The minimum value for an ILP problem is always greater than or equal to the minimum for the LP relaxation:

MIN for ILP ≥ MIN for LP relaxation

I cannot figure out why "MAX for ILP ≤ MAX for LP relaxation" and "MIN for ILP ≥ MIN for LP relaxation".

Can anyone explain, thx!

Answer 1

An ILP has an extra constraint than LP problem. The constraint is that all variables should be integers.

Hence, the optimal solution for an ILP shall be at best as good as an optimal solution for an LP problem, it can never be better.