Consider the binary knapsack polytope 18x1 + 3x2 + 20x3 + 17x4 + 6x5

+ 10x6 + 5x7 + 12x8 … 25 x1,c, x8 = 0 or 1

(a) Identify 5 minimal cover inequalities valid for this knapsack polytope, including the one involvin the last 3 variables, and briefly justify why each meets the requirements of definition 12.48 .

(b) Explain how the valid inequality forms of

(a) remain valid, and could contribute to sharpening the LP relaxation, if the knapsack constraints above were only part a larger model with many constraints.