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Homework_10_Solutions

# Homework_10_Solutions - HOMEWORK 10 SOLUTIONS GRADERS...

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HOMEWORK 10 SOLUTIONS GRADERS: ANDREW LI (AAL2131, A-I), LINWEI LI (LL2713, J-Q), & LEO LIN (CL2892, R-Z) 1. Problem 1 10 pts Let y ij = 1 if node i colored j 0 otherwise , i = 1 , . . . , n, j = 1 , 2 , 3 minimize 0 subject to: y i 1 + y i 2 + y i 3 = 1 , i = 1 , . . . , n y ij + y kj 1 j = 1 , 2 , 3 , ( i, k ) E y ij ∈ { 0 , 1 } ∀ i = 1 , . . . , n, j = 1 , 2 , 3 2. Problem 2 10 pts maximize y 1 + · · · + y n subject to: y i + y j 1 , ( i, j ) / E y i ∈ { 0 , 1 } , i = 1 , . . . , n 3. Problem 3 (a) 4 pts maximize 2 x 11 + 4 x 12 + 8 x 13 + 16 x 14 + 5 x 21 + 10 x 22 + 20 x 23 + 40 x 24 subject to: x 11 + 2 x 12 + 4 x 13 + 8 x 14 + x 21 + 2 x 22 + 4 x 23 + 8 x 24 15 - x 11 - 2 x 12 - 4 x 13 - 8 x 14 + x 21 + 2 x 22 + 4 x 23 + 8 x 24 2 x 11 + 2 x 12 + 4 x 13 + 8 x 14 - x 21 - 2 x 22 - 4 x 23 - 8 x 24 2 x 11 + 2 x 12 + 4 x 13 + 8 x 14 + x 21 + 2 x 22 + 4 x 23 + 8 x 24 2 x ij ∈ { 0 , 1 } , i = 1 , 2 , j = 1 , 2 , 3 , 4 (b) 2 pts minimize - 2 x 11 - 4 x 12 - 8 x 13 - 16 x 14 - 5 x 21 - 10 x 22 - 20 x 23 - 40 x 24 subject to: x 11 + 2 x 12 + 4 x 13 + 8 x 14 + x 21 + 2 x 22 + 4 x 23 + 8 x 24 15 - x 11 - 2 x 12 - 4 x 13 - 8 x 14 + x 21 + 2 x 22 + 4 x 23 + 8 x 24 2 - x 11 - 2 x 12 - 4 x 13 - 8 x 14 + x 21 + 2 x 22 + 4 x 23 + 8 x 24 ≤ - 2 - x 11 - 2 x 12 - 4 x 13 - 8 x 14 - x 21 - 2 x 22 - 4 x 23 - 8 x 24 ≤ - 2 x ij ∈ { 0 , 1 } , i = 1 , 2 , j = 1 , 2 , 3 , 4 (c) 4 pts 1

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2 GRADERS: ANDREW LI (AAL2131, A-I), LINWEI LI (LL2713, J-Q), & LEO LIN (CL2892, R-Z) Let x 0 ij = 1 - x ij minimize 2 x 0 11 + 4 x 0 12 + 8 x 0 13 + 16 x 0 14 + 5 x 0 21 + 10 x 0 22 + 20 x 0 23 + 40 x 0 24 subject to: - x 0 11 - 2 x 0 12 - 4 x 0 13 - 8 x 0 14 - x 0 21 - 2 x 0 22 - 4 x 0 23 - 8 x 0 24 ≤ - 15 x 0 11 + 2 x 0 12 + 4 x 0 13 + 8 x 0 14 - x 0 21 - 2 x 0 22 - 4 x 0 23 - 8 x 0 24 2 x 0 11 + 2 x 0 12 + 4 x 0 13
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