Given a flow network G = (V, E) and any flow between s and t, let (A,B) be a minimum cut. Then, what's the relation between the maximum possible net flow between s and t, and the capacity of the cut (A, B). (a) Greater than (b) Less than (c) Equal to (d) Can't determine the relation
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- Given a flow network as below with S and T as source and sink (destination). The pair of integers on each edge corresponds to the flow value and the capacity of that edge. For instance, the edge (S.A) has capacity 16 and currently is assigned a flow of 5 (units). Assume that we are using the Ford-Fullkerson's method to find a maximum flow for this problem. Fill in the blanks below with your answers. a) An augmenting path in the corresponding residual network is Note: give you answer by listing the vertices along the path, starting with S and ending with T, e.g., SADT (note that this is for demonstration purpose only and may not be a valid answer), with no spaces or punctuation marks, i.e., no commas "," or full stops ".". If there are more than one augmenting path, then you can choose one arbitrarily. b) The maximum increase of the flow value that can be applied along the augmenting path identified in Part a) is c) The value of a maximum flow is Note: your answers for Part b) and Part…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.Consider the flow network shown in the following figure (left), where the label next to each arc is its capacity, and the initial s-t flow on right. (b) Apply the Ford-Fulkerson algorithm to N starting from the flow f. The augmenting path used in each step of the algorithm must be given. (c) Give the cut that corresponds to the maximum flow obtained in (b), which is suggested in the Max-Flow-Min-Cut theorom.
- 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.3 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.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.
- Question 1Draw the residual network obtained from this flow. Question2Perform 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. Question 3Exhibit 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.4. Find the maximum flow from source (node 0) to destination (node 5) from the following flow graph. Show the residual network at each step. 12 1 16 20 10||4 7 13 4 14 4. 2.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 a network be given by nodes V = {S, A, B,C, D,T} and arcs, capacities e and a flow f according to the table: SA SC AB Bт АC DC BD DT AT CT 3 ? 4 1 1 ? 3 4 3 5 ? 3 9. 3 5 1 1 (a) Complete f and e such that f is an admissible flow from S to T. Justify your answer. What is the value of the flow?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;