Decide whether you think the following state is true or false. If it is true, give a short explanation. If it is false, give a counter example. Let G be an arbitrary flow network, with a source s, a sink t, and a positive integer capacity ce on every edge. If ƒ is a maximum s –t flow in G, then f saturates every edge out of s with flow (e.g., for all edges e out of s, we have f(e) = c.).

Decide whether you think the following state is true or false. If it is true, give a short explanation. If it is false, give a counter example. Let G be an arbitrary flow network, with a source s, a sink t, and a positive integer capacity ce on every edge. If ƒ is a maximum s –t flow in G, then f saturates every edge out of s with flow (e.g., for all edges e out of s, we have f(e) = c.).

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

See similar textbooks

Similar questions

1. Recall that a flow network is a directed graph G = (V, E) with a source s, a sink t, and a capacity function c: V x V + Rj that is positive on E and 0 outside E.We only consider finite graphs here. Also, note that every flow network has a maximum flow. This sounds obvious but requires a proof (and we did not prove it in the video lecture). Which of the following statements are true for all flow networks (G, s,t,c)? O 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 x V R. That is, if f(u, v) # f'(u, v) for some u, ve V. 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 0 then f(u, v) = 0 for all u, v. The number of maximum flows is 1 or infinity. The number of minimum cuts is finite. Need help, as you can see the checked boxes is not the right answer, something is missing…
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).
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.
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.
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 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'
Let f be an s,t-flow in an s, t-network D = (V, A) with capacities c : A → R>o, and assume that there is no augmenting path. Let X be the set of vertices that can be reached from s by unsaturated paths. Let (a, b) E A(X,V\ X). Explain why f(a,b) = c(a, b).
For a directed graph G = (V,E) (source and sink in V denoted by s and t respec- tively) with capacities c: E→+, and a flow f: E→, the support of the flow f on G is the set of edges E:= {e E| f(e) > 0}, i.e. the edges on which the flow function is positive. Show that for any directed graph G = (V,E) with non-negative capacities e: Ethere always exists a maximum flow f*: E→+ whose support has no directed cycle.
Circle T(rue) or F(alse). T F (a) integer capacity ce on 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). Let G be an arbitrary flow network, with a source s, a sink t, and a positive - T F (b) Then it is possible that the residual graph contains (v, u), but not (u, v). Let (u, v) be an edge in a maximum flow graph with capacity greater than 0.
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.
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.