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CSE, IIT KGP Application of Network Flows: Matrix Rounding
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CSE, IIT KGP Matrix Rounding 16.5 13.5 10.9 9.8 18.5 14.4 Ʃ Ʃ [16,17] [14,15] [10,11] [18,19] [9,10] [13,14] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3
![CSE, IIT KGP Matrix Rounding Ʃ Ʃ [16,17] [14,15] [10,11] [18,19] [9,10] [13,14] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3](http://images.slideplayer.com/27/9104281/slides/slide_2.jpg "CSE, IIT KGP Matrix Rounding Ʃ Ʃ [16,17] [14,15] [10,11] [18,19] [9,10] [13,14] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3")
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CSE, IIT KGP Solution Step 1: Represent the consistent rounding problem as a feasible flow problem.Step 1: Represent the consistent rounding problem as a feasible flow problem. Step 2: Convert the feasible flow problem into a feasible circulation problem.Step 2: Convert the feasible flow problem into a feasible circulation problem. Step 3: Convert the feasible circulation problem into a Demand and Supplies problem.Step 3: Convert the feasible circulation problem into a Demand and Supplies problem. Step 4: Convert the transportation problem into a network flows problem.Step 4: Convert the transportation problem into a network flows problem.
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CSE, IIT KGP Solution Step 1: Represent the consistent rounding problem as a feasible flow problem.Step 1: Represent the consistent rounding problem as a feasible flow problem. Step 2: Convert the feasible flow problem into a feasible circulation problem.Step 2: Convert the feasible flow problem into a feasible circulation problem. Step 3: Convert the feasible circulation problem into a Demand and Supplies problem.Step 3: Convert the feasible circulation problem into a Demand and Supplies problem. Step 4: Convert the transportation problem into a network flows problem.Step 4: Convert the transportation problem into a network flows problem.
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CSE, IIT KGP [16,17] [14,15] [10,11] [18,19] [9,10] [13,14] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] Feasible Flow Problem Formulation
![CSE, IIT KGP [16,17] [14,15] [10,11] [18,19] [9,10] [13,14] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] Feasible Flow Problem Formulation](http://images.slideplayer.com/27/9104281/slides/slide_5.jpg "CSE, IIT KGP [16,17] [14,15] [10,11] [18,19] [9,10] [13,14] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] Feasible Flow Problem Formulation")
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CSE, IIT KGP Solution Step 1: Represent the consistent rounding problem as a feasible flow problem.Step 1: Represent the consistent rounding problem as a feasible flow problem. Step 2: Convert the feasible flow problem into a feasible circulation problem.Step 2: Convert the feasible flow problem into a feasible circulation problem. Step 3: Convert the feasible circulation problem into a Demand and Supplies problem.Step 3: Convert the feasible circulation problem into a Demand and Supplies problem. Step 4: Convert the transportation problem into a network flows problem.Step 4: Convert the transportation problem into a network flows problem.
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CSE, IIT KGP [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] Feasible Circulation Problem [0,∞] Add an edge from t → s with bounds [0,∞]
![CSE, IIT KGP [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] Feasible Circulation Problem [0,∞] Add an edge from t → s with bounds [0,∞]](http://images.slideplayer.com/27/9104281/slides/slide_7.jpg "CSE, IIT KGP [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] Feasible Circulation Problem [0,∞] Add an edge from t → s with bounds [0,∞]")
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CSE, IIT KGP Solution Step 1: Represent the consistent rounding problem as a feasible flow problem.Step 1: Represent the consistent rounding problem as a feasible flow problem. Step 2: Convert the feasible flow problem into a feasible circulation problem.Step 2: Convert the feasible flow problem into a feasible circulation problem. Step 3: Convert the feasible circulation problem into a Demand and Supplies problem.Step 3: Convert the feasible circulation problem into a Demand and Supplies problem. Step 4: Convert the transportation problem into a network flows problem.Step 4: Convert the transportation problem into a network flows problem.
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CSE, IIT KGP Transportation Problem [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] [0,∞] c(x 1,y 1 ) = 1 l - (x 1 ) = 18 l + (x 1 ) = 6+7+4=17 b(x 1 ) = 18-17 = 1
![CSE, IIT KGP Transportation Problem [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] [0,∞] c(x 1,y 1 ) = 1 l - (x 1 ) = 18 l + (x 1 ) = 6+7+4=17 b(x 1 ) = = 1](http://images.slideplayer.com/27/9104281/slides/slide_9.jpg "CSE, IIT KGP Transportation Problem [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] [0,∞] c(x 1,y 1 ) = 1 l - (x 1 ) = 18 l + (x 1 ) = 6+7+4=17 b(x 1 ) = = 1")
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CSE, IIT KGP vl - (v)l + (v)b(v)=l - (v)-l + (v) s040-40 (D) x1x1 1817+1 (S) x2x2 98 x3x3 1312+1 (S) y1y1 1516-1 (D) y2y2 1314-1 (D) y3y3 910-1 (D) t400+40 (S) [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] [0,∞] S – Supply Node D – Demand Node
![CSE, IIT KGP vl - (v)l + (v)b(v)=l - (v)-l + (v) s (D) x1x (S) x2x2 98 x3x (S) y1y (D) y2y (D) y3y (D) t (S) [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] [0,∞] S – Supply Node D – Demand Node](http://images.slideplayer.com/27/9104281/slides/slide_10.jpg "CSE, IIT KGP vl - (v)l + (v)b(v)=l - (v)-l + (v) s (D) x1x (S) x2x2 98 x3x (S) y1y (D) y2y (D) y3y (D) t (S) [18,19] [9,10] [13,14] [16,17] [14,15] [10,11] x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 st [6,7] [3,4] [7,8] [4,5] [2,3] [5,6] [2,3] [4,5] [0,∞] S – Supply Node D – Demand Node")
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CSE, IIT KGP 1 1 1 1 1 1 x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 s t 1 1 1 1 1 1 1 1 1 Back to a Network Flows Problem ∞ s' t' If b(v) ≥ 0, add an edge from s’ to v else add an edge from v to t’, with capacity b(v). Add a new source s’ and a new sink t’. 1 1 1 1 1 1 40 Is there a flow equal to the total capacity on the edges leaving s’? (In this case 43)
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