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Tutorial11solution - Tutorial 11 Outline of Solutions Q1 In...

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Tutorial 11: Outline of Solutions Q1. In the following two maximum flow problems with source s = 1 and sink t = 5, determine all s - t cuts. Find the minimum s - t cut in each case. Solution : (8,0) (4,0) (2,0) (6,0) (4,6) 2 4 5 3 (4,4) 1 30 25 15 20 12 2 3 5 4 18 22 12 1 (0,2) (4,0) u ji u ij i j (u ij, u ji ) i j i j For i < j u ij (a) Minimum s-t cut is S = { 1 } , S = { 2 , 3 , 4 , 5 } with capacity 37. S S ( S, S ) u ( S, S ) { 1 } { 2,3,4,5 } { (1 , 2) , (1 , 3) } 37 { 1,2 } { 3,4,5 } { (1 , 3) , (2 , 3) , (2 , 4) , (2 , 5) } 77 { 1,3 } { 2,4,5 } { (1 , 2) , (3 , 4) , (3 , 5) } 59 { 1,4 } { 2,3,5 } { (1 , 2) , (1 , 3) , (4 , 5) } 55 { 1,2,3 } { 4,5 } { (2 , 4) , (2 , 5) , (3 , 4) , (3 , 5) } 84 { 1,2,4 } { 3,5 } { (1 , 3) , (2 , 3) , (2 , 5) , (4 , 5) } 65 { 1,3,4 } { 2,5 } { (1 , 2) , (3 , 5) , (4 , 5) } 65 { 1,2,3,4 } { 5 } { (2 , 5) , (3 , 5) , (4 , 5) } 60 (b) Convert the graph into a directed graph and drop arcs with zero capacity. Minimum s-t cuts are S = { 1 } , S = { 2 , 3 , 4 , 5 } and S = { 1 , 3 } , S = { 2 , 4 , 5 } with capacity 10. 1
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8 4 2 6 4 6 2 4 5 3 4 4 1 4 2 S S ( S, S ) u ( S, S ) { 1 } { 2,3,4,5 } { (1 , 2) , (1 , 3) , (1 , 4) } 10 { 1,2 } { 3,4,5 } { (1 , 4) , (1 , 3) , (2 , 3) , (2 , 5) } 20 { 1,3 } { 2,4,5 } { (1 , 2) , (1 , 4) , (3 , 5) } 10 { 1,4 } { 2,3,5 } { (1 , 2) , (1 , 3) , (4 , 3) , (4 , 5) } 14 { 1,2,3 } { 4,5 } { (1 , 4) , (2 , 5) , (3 , 5) } 12 { 1,2,4 } { 3,5 } { (1 , 3) , (2 , 3) , (2 , 5) , (4 , 3) , (4 , 5) } 24 { 1,3,4 } { 2,5 } { (1 , 2) , (3 , 5) , (4 , 5) } 12 { 1,2,3,4 } { 5 } { (2 , 5) , (3 , 5) , (4 , 5) } 14 Q2. Consider the maximum flow problem between source 1 and sink 4 in the network. u ij i j 1 2 4 3 2 2 4 1 6 (a) Write the linear program for the maximum flow problem. (b) Find the optimal arc flows by intuition. Is this solution a basic solution? (c) Write the dual linear program. Use complementary slackness to determine a dual optimal solution and identify the minimum s - t cut based on this. 2
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Solution : (a) Maximum flow problem is formulated as the LP: max v s.t. x 12 + x 13 - v = 0 - x 12 - x 32 + x 24 = 0 - x 13 + x 32 + x 34 = 0 - x 24 - x 34 + v = 0 0 x 12 2 0 x 13 6 0 x 24 4 0 x 32 1 0 x 34 2 (b) Optimal solution is v * = 5 , x * 12 = 2 , x * 13 = 3 , x * 32 = 1 , x * 24 = 3 , x * 34 = 2. This is a basic solution since, there are 6 linearly independent active constraints at this solution.
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