This is an optimization proof-based question. Please write it in detail. Throughout assume that G = (V, E) is a directed graph with |V| = n and |E| = m unlessotherwise stated. Assume that all edge capacities are integral and that there are no directed2-cycles. Let (G, u, s, t) be a network and f* a maximum flow. Consider the following variation ofFord-Fulkerson.(1) Set f(e) = 0 for all e E E.(2) While there exists an s-t path in Gƒ:(3)Compute the s-t path P in G that maximizes y := {uf(e)| e E P}.(4)Augment f along P by y.Let fo, f1, ... , fT be the sequence of flows that the algorithm computes (where) = value(fo) < value(f1) < ··… < value(fr) = value(f*)). Hint: The previous problemshould help with (a) and (b) below.(a) Prove that value(fi) > 1 value(f*).(b) Prove that value(fi) > value(fi-1) + ± (value(f*) – value(fi-1)) for all i E [T].m(c) Prove that value(f;) > value(f*) (1 – (1 – ±)') for all i € [T].(d) Prove that T < m In(2 value(f*)). (In other words, this is an upper bound for thenumber of iterations of this algorithm.)

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Description: This is an optimization proof-based question. Please write it in detail. Throughout assume that G = (V, E) is a directed graph with |V| = n and |E| = m unlessotherwise stated. Assume that all edge capacities are integral and that there are no directe

Let G be a directed graph. G=(V,E) with |V|=n and |E|=m Then |V|=n=6, |E|=m=9 Now, we…

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Let fo, f1,.…, fr be the sequence of flows that the algorithm computes (where 0 = value(fo) < value(f1) < ·…< value(fr) = value(f*)). Hint: The previous problem should help with (a) and (b) below. (a) Prove that value(f1) > - value(f*). (b) Prove that value(f;) > value(fi-1) + ± (value(f*) – value(ƒ;-1)) for all i e [T]. m (c) Prove that value(f;) > value(f*) (1– (1 – ±)') for all i e [T].

Let fo, f1,.…, fr be the sequence of flows that the algorithm computes (where 0 = value(fo) < value(f1) < ·…< value(fr) = value(f)). Hint: The previous problem should help with (a) and (b) below. (a) Prove that value(f1) > - value(f). (b) Prove that value(f;) > value(fi-1) + ± (value(f) – value(ƒ;-1)) for all i e [T]. m (c) Prove that value(f;) > value(f) (1– (1 – ±)') for all i e [T].

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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This is an optimization proof-based question. Please write it in detail.

Throughout assume that G = (V, E) is a directed graph with |V| = n and |E| = m unless
otherwise stated. Assume that all edge capacities are integral and that there are no directed
2-cycles.

Let (G, u, s, t) be a network and f* a maximum flow. Consider the following variation of
Ford-Fulkerson.
(1) Set f(e) = 0 for all e E E.
(2) While there exists an s-t path in Gƒ:
(3)
Compute the s-t path P in G that maximizes y := {uf(e)| e E P}.
(4)
Augment f along P by y.
Let fo, f1, ... , fT be the sequence of flows that the algorithm computes (where
) = value(fo) < value(f1) < ··… < value(fr) = value(f*)). Hint: The previous problem
should help with (a) and (b) below.
(a) Prove that value(fi) > 1 value(f*).
(b) Prove that value(fi) > value(fi-1) + ± (value(f*) – value(fi-1)) for all i E [T].
m
(c) Prove that value(f;) > value(f*) (1 – (1 – ±)') for all i € [T].
(d) Prove that T < m In(2 value(f*)). (In other words, this is an upper bound for the
number of iterations of this algorithm.)

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