If all the capacities in the given network are integers, prove that the algorithm always returns a solution in which the flow through each edge is an integer. For some applications, this fact is crucial.
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If all the capacities in the given network are integers, prove that the
algorithm always returns a solution in which the flow through each edge is an integer.
For some applications, this fact is crucial.
Step by step
Solved in 2 steps
- 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.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.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.
- 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.3. Given an arbitrary network with integer edge capacities, show that the max flow f exists such that f(a) is an integer for all a E ALet 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.
- 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.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.
- 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.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. 3 2 b 6 Fig. 1 A flow network, with sources and sink t. The numbers next to the edges are the capacities.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) ?