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arc. Consider the network of Figure 2, where the capacities of arcs are given in rectangles at each (i) Knowing that (W, W) with W = network. {s, a, b, c} is a minimal s- t cut suggest a maximal flow for this 100 80 60 50 70 30 50 b 40 d 80 100 Figure 2: Network to be considered in Combinatorial Optimisation Problem 3 Hints: The flow can be suggested by copying the network of Figure 2 and indicating the flow through each arc as done on Figure 1. The flow can be found by using the relations between maximal flows and minimal cuts and without applying the Ford-Fulkerson algorithm. (ii) Justify that the suggested flow is maximal by running one iteration of the Ford-Fulkerson algorithm ending with nonbreakthrough. Show your working. (iii) Suppose that f is a maximal flow through the network. Find all arcs e for which the value f(e) is the same for all maximal flows, and all arcs e for which it can be different in a different maximal flow. Justify. (iv) Suppose that we are required to increase the value of the maximal flow by 30, and that we need to achieve this by increasing the capacity of just one arc. Which arc (or arcs) can be used for this task? For any such arc that you find, draw the resulting network and indicate the maximal flow. For the remaining arcs, give your argument without finding any maximal flows why these arcs cannot be used for this task.

arc. Consider the network of Figure 2, where the capacities of arcs are given in rectangles at each (i) Knowing that (W, W) with W = network. {s, a, b, c} is a minimal s- t cut suggest a maximal flow for this 100 80 60 50 70 30 50 b 40 d 80 100 Figure 2: Network to be considered in Combinatorial Optimisation Problem 3 Hints: The flow can be suggested by copying the network of Figure 2 and indicating the flow through each arc as done on Figure 1. The flow can be found by using the relations between maximal flows and minimal cuts and without applying the Ford-Fulkerson algorithm. (ii) Justify that the suggested flow is maximal by running one iteration of the Ford-Fulkerson algorithm ending with nonbreakthrough. Show your working. (iii) Suppose that f is a maximal flow through the network. Find all arcs e for which the value f(e) is the same for all maximal flows, and all arcs e for which it can be different in a different maximal flow. Justify. (iv) Suppose that we are required to increase the value of the maximal flow by 30, and that we need to achieve this by increasing the capacity of just one arc. Which arc (or arcs) can be used for this task? For any such arc that you find, draw the resulting network and indicate the maximal flow. For the remaining arcs, give your argument without finding any maximal flows why these arcs cannot be used for this task.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 10E
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arc. Consider the network of Figure 2, where the capacities of arcs are given in rectangles at each (i) Knowing that (W, W) with W = network. {s, a, b, c} is a minimal s- t cut suggest a maximal flow for this
Transcribed Image Text:arc. Consider the network of Figure 2, where the capacities of arcs are given in rectangles at each (i) Knowing that (W, W) with W = network. {s, a, b, c} is a minimal s- t cut suggest a maximal flow for this
100 80 60 50 70 30 50 b 40 d 80 100 Figure 2: Network to be considered in Combinatorial Optimisation Problem 3 Hints: The flow can be suggested by copying the network of Figure 2 and indicating the flow through each arc as done on Figure 1. The flow can be found by using the relations between maximal flows and minimal cuts and without applying the Ford-Fulkerson algorithm. (ii) Justify that the suggested flow is maximal by running one iteration of the Ford-Fulkerson algorithm ending with nonbreakthrough. Show your working. (iii) Suppose that f is a maximal flow through the network. Find all arcs e for which the value f(e) is the same for all maximal flows, and all arcs e for which it can be different in a different maximal flow. Justify. (iv) Suppose that we are required to increase the value of the maximal flow by 30, and that we need to achieve this by increasing the capacity of just one arc. Which arc (or arcs) can be used for this task? For any such arc that you find, draw the resulting network and indicate the maximal flow. For the remaining arcs, give your argument without finding any maximal flows why these arcs cannot be used for this task.
Transcribed Image Text:100 80 60 50 70 30 50 b 40 d 80 100 Figure 2: Network to be considered in Combinatorial Optimisation Problem 3 Hints: The flow can be suggested by copying the network of Figure 2 and indicating the flow through each arc as done on Figure 1. The flow can be found by using the relations between maximal flows and minimal cuts and without applying the Ford-Fulkerson algorithm. (ii) Justify that the suggested flow is maximal by running one iteration of the Ford-Fulkerson algorithm ending with nonbreakthrough. Show your working. (iii) Suppose that f is a maximal flow through the network. Find all arcs e for which the value f(e) is the same for all maximal flows, and all arcs e for which it can be different in a different maximal flow. Justify. (iv) Suppose that we are required to increase the value of the maximal flow by 30, and that we need to achieve this by increasing the capacity of just one arc. Which arc (or arcs) can be used for this task? For any such arc that you find, draw the resulting network and indicate the maximal flow. For the remaining arcs, give your argument without finding any maximal flows why these arcs cannot be used for this task.
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