Final answer:
The statement is true. If the LP relaxation's optimal value is 100, any integer feasible solution with an objective function value of 95 or more is within 5% of the global optimal integer solution exactly because the LP relaxation sets an upper bound for the ILP solution.
Step-by-step explanation:
The question relates to the field of operations research specifically to Integer Linear Programming (ILP) and Linear Programming (LP) relaxations. The statement addresses the proximity of any integer feasible solution to the global optimal solution, given that the LP relaxation has an optimal objective value. It asks if an integer feasible solution with a value of 95 or more, which is a 5% deviation from the LP relaxation's optimal value of 100, is guaranteed to be within 5% of the global optimal integer solution.
This statement is indeed true. If the LP relaxation has an optimal value of 100, this sets an upper bound for the ILP since integer solutions are more constrained. An integer feasible solution with a value of 95 is only 5 units less than the relaxed solution. It means this integer solution is at least 95% efficient with respect to the LP relaxation's bound. Therefore, it must be within 5% of the global optimal integer solution because the LP relaxation provides the best possible objective value without the integer constraints and any integer solution cannot exceed this bound.