Prove that the method always produces a solution in which the flow through each edge is an integer if all the capacities in the provided network are integers. This fact is essential for several applications.
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Prove that the method always produces a solution in which the flow through each edge is an integer if all the capacities in the provided network are integers.
This fact is essential for several applications.
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- Prove that if all of the capacities in the given network are integers, the method always yields a solution with an integer flow across each edge.This is important for some applications.If all the capacities in the given network are integers, prove that thealgorithm always returns a solution in which the flow through each edge is an integer.For some applications, this fact is crucial.True or false: For any flow network G and any maximum flow on G, there is always an edge e such that increasing the capacity of e increases the maximum flow of the network. Justify your answer. Doğru veya yanlış: Herhangi bir akış ağı G ve G üzerindeki herhangi bir maksimum akış için, her zaman bir e kenarı vardır, öyle ki e'nin kapasitesini artırmak ağın maksimum akışını arttırır. Cevabınızı gerekçelendirin.
- Consider a volleyball net that consists of a mesh with m squares on the horizontal dimension and n squares on the vertical. What is the maximum number of strings that can be cut before the net falls apart into two pieces? Solve the problem as a network problem.Question 10. The graph below represents a network and the capacities are the number written on edges. The source is node a, and the target is node e. We use the Ford-Fulkerson method to find the max flow. The questions below are about the first iteration of the method. a d 1 3 7 b e 1 2 1 (a) Indicate one augmenting path. (b) How much flow can be pushed on the path you indicated at (a)? (c) Draw the residual graph Gƒ for the flow you have indicated at (b). Question 8. A com Show that hLet G= (V, E) be an arbitrary flow network with source s and sink t, and a positive integer capacity c(u, v) for each edge (u, v)∈E. Let us call a flow even if the flow in each edge is an even number. Suppose all capacities of edges in G are even numbers. Then,G has a maximum flow with an even flow value.
- 0/16 0/12 0/8 0/4 22 0/8 0/5 0/11 VA 0/13 0/14 V5 0/2 0/11 0/10 Each edge is annotated with the current flow (initially zero) and the edge's capacity. In general, a flow of x along an edge with capacity y is shown as x/y. (a) Show the residual graph that will be created from this network with the given (empty) flow. In drawing a residual graph, to show a forward edge with capacity x and a backward edge with capacity y, annotate the original edge *; y. (b) What is the bottleneck edge of the path (S, V₁, V3, V5, t) in the residual graph you have given in answer to part (a) ? (c) Show the network with the flow (s, V₁, V3, V5, t) that results from augmenting the flow based on the path of the residual graph you have given in answer to part (a). (d) Show the residual graph for the network flow given in answer to part (c). (e) What is the bottleneck edge of the path (s, v3, v4, t) in the residual graph you have given in answer to part (d) ?True or False Let G be an arbitrary flow network, with a source s, a sink t, and a positiveinteger capacity ceon every edge e. If f is a maximum s −t flow in G, then f saturates every edge out of s with flow (i.e., for all edges e out of s, we have f (e) = ce).Compute the maximum flow in the following flow network using Ford-Fulkerson's algorithm. 12 8 14 15 11 2 5 1 3 d 7 f 6 What is the maximum flow value? A minimum cut has exactly which vertices on one side? 4.
- How can we merge parallel edges that are in the same direction in a flow network G?Problem 2. Find the maximum flow in the flow network shown in figure 1. In the flow network ‘s’ is the source vertex and ‘t’ is the destination vertex. The capacity of each of the edges are given in the figure.One can manually count path lengths in a graph using adjacency matrices. Using the simple example below, produces the following adjacency matrix: A B A 1 1 B 1 0 This matrix means that given two vertices A and B in the graph above, there is a connection from A back to itself, and a two-way connection from A to B. To count the number of paths of length one, or direct connections in the graph, all one must do is count the number of 1s in the graph, three in this case, represented in letter notation as AA, AB, and BA. AA means that the connection starts and ends at A, AB means it starts at A and ends at B, and so on. However, counting the number of two-hop paths is a little more involved. The possibilities are AAA, ABA, and BAB, AAB, and BAA, making a total of five 2-hop paths. The 3-hop paths starting from A would be AAAA, AAAB, AABA, ABAA, and ABAB. Starting from B, the 3-hop paths are BAAA, BAAB, and BABA. Altogether, that would be eight 3-hop paths within this graph. Write a program…