Suppose that G = (V, E) is a directed graph with |V| n and JE] = m unlesS otherwise stated. Also suppose that all edge capacities are integral and that there are no directed 2-cycles. Let (G, u, s, t) be a network with a flow f : E → R>o. Let ƒ* be a maximum flow. (Recall that G¡ = (V, E¡) is the residual graph for f.) %3D (a) Let S C V with s € S and t ¢ S, and let 87 (S) = {(i, j) E E¡ | i E S and j ¢ S}. Show that %3D value(f*) < value(f)+ £ us(e) eɛ8;(S) where uf(e) is the residual capacity of e. (b) Let y> 0 be the maximum value so that there is an s-t path P in G¡ with uf(e) > y for all e e P. Show that value(f*) < value(f)+ym.
Suppose that G = (V, E) is a directed graph with |V| n and JE] = m unlesS otherwise stated. Also suppose that all edge capacities are integral and that there are no directed 2-cycles. Let (G, u, s, t) be a network with a flow f : E → R>o. Let ƒ* be a maximum flow. (Recall that G¡ = (V, E¡) is the residual graph for f.) %3D (a) Let S C V with s € S and t ¢ S, and let 87 (S) = {(i, j) E E¡ | i E S and j ¢ S}. Show that %3D value(f*) < value(f)+ £ us(e) eɛ8;(S) where uf(e) is the residual capacity of e. (b) Let y> 0 be the maximum value so that there is an s-t path P in G¡ with uf(e) > y for all e e P. Show that value(f*) < value(f)+ym.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Suppose that G = (V, E) is a directed graph with |V| = n and |E| = m unless otherwise stated.
Also suppose that all edge capacities are integral and that there are no directed 2-cycles.
Let (G, u, s, t) be a network with a flow f : E → R>o. Let f* be a maximum flow. (Recall
that Gf = (V, E;) is the residual graph for f.)
(a) Let S C V with s E S and t ¢ S, and let 8 (S) = {(i, j) e E¡ | i E S and j ¢ S}. Show
that
value(f*) < value(f)+ £ us(e)
eE8;(S)
where uf(e) is the residual capacity of e.
(b) Let y > 0 be the maximum value so that there is an s-t path P in Gƒ with uf(e) > y
for all e e P. Show that value(f*) < value(f)+ym.
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