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.

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.

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 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.
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.
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.
Show the network with the flow that results from augmenting the flow based on the path (?, ?2, ?3, ?1, ?4, ?) of the residual graph you have given in answer to part. Show the residual graph for the network flow given in answer to part. 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 (i), and augments flow along the edges (?,?1,?3,?) and (?,?2,?5,?).
Consider the following directed network with flows written as the first number and edge capacity as the second on each edge: Part 1 Draw the residual network obtained from this flow. Part 2 Perform two steps of the Ford Fulkerson algorithm on this network, each using the residual graph of the cumulative flow, and the augmenting paths and flow amounts specified below. After each augment, draw two graphs, preferably side by side; these are graphs of: a) The flow values on the edges b) Residual network The augmenting paths and flow amounts are: i) s → b→d c→t with flow amount 7 Units. ii) s → b→ c→ t with 4 units. Note for continuity your second graph should be coming from the one in (i) NOT from the initial graph. Part 3 Exhibit a maximum flow with flow values on the edges, state its value, and exhibit a cut (specified as a set of vertices) with the same value.
Let f be a flow of flow network G and f' a flow of residual network Gf . Show that f +f' is a flow of G.
Lets say there is a flow network called G and a flow in G called f, we say that f saturates an edge e if the flow value on that edge is equal to its capacity. The flow sat problem is: given a flow network G and a positive integer k, determine if there exists a flow f in G such that f saturates at least k edges of G. Prove that Flow sat is NP-complete.
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.
Describe how to construct an incremental network in the Ford-Fulkerson algorithm in order to find the maximal flow through a network flow model with minimal overall cost.
Prove that the network flow algorithm presented in this section maintains the loop invariant that it always holds a legal flow. Do this by proving that the changes to the flow do not violate any edge capacities or create leaks at nodes. Also prove that progress is made because the total flow increases. You need to be careful with your plus and minus signs.
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.