Consider the following flow in the s, t-network D. 5,1 3,0 10,1 10,5 6,2 7,4 4,3 8,6 2,1 3,1 4,2 (a) Verify that the given flow is feasible. (b) Use the FFEK algorithm to find a maximum flow of D starting with the given flow. Prove that your final flow is maximum by giving a minimum cut. At every step of your algorithm, list the augmenting path used.

Consider the following flow in the s, t-network D. 5,1 3,0 10,1 10,5 6,2 7,4 4,3 8,6 2,1 3,1 4,2 (a) Verify that the given flow is feasible. (b) Use the FFEK algorithm to find a maximum flow of D starting with the given flow. Prove that your final flow is maximum by giving a minimum cut. At every step of your algorithm, list the augmenting path used.

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter8: Network Models

Section8.3: Maximum-flow Problems

Problem 11P

See similar textbooks

Similar questions

Let G=(V,E) be a flow network and suppose that you are given a cut (S,T) of minimum capacity x. What else can be said about G? a. The maximum flow of G is x b. There are no augmenting paths in c. All valid cuts of G have capacity x d. A and B e. All of the above. G, the residual network of G 'ƒ'
Apply the shortest-augmenting path algorithm to find a maximum flow in the following networks.
Let G = (V, E) be a flow network with source s and sink t. We say that an edge e is a bottleneck in G if it belongs to every minimum capacity cut separating s from t. Give a polynomial-time algorithm to determine if a given edge e is a bottleneck in G.
Show the final flow that the Ford-Fulkerson Algorithm finds for this network, given that it proceeds to completion from the flow rates you have given in your answer to part, and augments flow along the edges (?,?1,?3,?) and (?,?2,?5,?). Identify a cut of the network that has a cut capacity equal to the maximum flow of the network.
Apply the augmenting algorithm to the following flow 4/6 В D 8/8 6/8 3/3 A 1/6 1/7 1/3 3/10 5/9 E 3/3 in order to find (a) a maximal flow in the network; (b) the corresponding minimal cut;
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.
Only considering Finite graphs, also note that every flow network has a maximum flow. Which of the following statements are true for all flow networks (G, s, t, c)? • IfG = (V, E) has as cycle then it has at least two different maximum flows. (Recall: two flows f, f' are different if they are different as functions V × V -> R. That is, if f (u, u) + f' (u, v) for some u, v EV. The number of maximum flows is at most the number of minimum cuts. The number of maximum flows is at least the number of minimum cuts. If the value of f is O then f(u, v) = O forallu, U. | The number of maximum flows is 1 or infinity. The number of minimum cuts is finite.
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).
Show the residual graph for the network flow given in answer to part (a) Show the final flow that the Ford-Fulkerson Algorithm finds for this network, given that it proceeds to completion from the flow rates you have given in your answer to part (a), and augments flow along the edges (?,?1,?3,?) and (?,?2,?5,?). Identify a cut of the network that has a cut capacity equal to the maximum flow of the network.
Consider the flow network G shown in figure 1 with source s and sink t. The edge capacities are the numbers given near each edge. (a) Find a maximum flow in this network. Once you have done this, draw a copy of the original network G and clearly indicate the flow on each edge of G in your maximum flow. (b) Find a minimum s-t cut in the network, i.e. name the two (non-empty) sets of vertices that define a minimum cut. Also, say what its capacity is. g C 4 3 6 d b 3 Fig.1 A flow network, with sources and sink t. The numbers next to the edges are the capacities.
The picture below shows a network with capacities c(e) and an actual s-t- flow f. Show that the flow f is not maximal and, starting from f, compute a maximum flow and minimum cut S with the algorithm of Edmonds and Karp. Specify the cut edges of S. a 10/5 3/0 3/0 5/5 8/8 b 10/8 = 8/8 5/5 3/0 3/0 10/5 d
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 h