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Digraphs+Theory,+Algorithms+and+Applications_Part10

# Digraphs+Theory,+Algorithms+and+Applications_Part10 - 3.13...

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3.13 Exercises 163 do augment x along P ; 6. C := C/ 2 7. return x Prove that the algorithm correctly determines a maximum flow in the input network N . (d) Argue that every time Step 4 is performed the residual capacity of every minimum ( s, t )-cut is at most 2 C | A | . (e) Argue that the number of augmentations performed in Step 5 is at most O ( | A | ) before Step 6 is executed again. (f) Conclude that Max-flow by scaling can be implemented so that its complexity becomes O ( | A | 2 log U ). Compare this complexity to that of other flow algorithms in this chapter. 3.27. Show how to find a maximum ( s, t )-flow in the network of Figure 3.24 using (a) The Ford-Fulkerson method. (b) Dinic’s algorithm. (c) The preflow-push algorithm. (d) The MKM-algorithm described in Exercise 3.25. (e) The scaling algorithm described in Exercise 3.26. 5 15 4 5 10 5 6 15 10 30 20 s t 9 25 Figure 3.24 A network with lower bounds and cost equal to zero on all arcs and capacities as indicated on the arcs. 3.28. (+) Rounding a real-valued flow . Let N = ( V, A, l, u ) be a network with source s and sink t and all data on the arcs non-negative integers (note that some of the lower bounds may be non-zero). Suppose x is a real-valued feasible flow in N such that x ij is a non-integer for at least one arc. (a) Prove that there exists a feasible integer flow x 0 in N with the property that | x ij - x 0 ij | < 1 for every arc ij A . (b) Suppose now that | x | is an integer. Prove that there exists an integer feasible flow x 00 in N such that | x 00 | = | x | . (c) Describe algorithms to find the flows x 0 , x 00 above. What is the best complexity you can achieve? 3.29. Finding a feasible circulation. Turn the proof of Theorem 3.8.2 into a polynomial algorithm which either finds a feasible circulation, or a proof that none exists. What is the complexity of the algorithm? 3.30. Residual networks of networks with non-zero lower bounds . Show how to modify the definition of x ˜ x in order to obtain an analogue of

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164 3. Flows in Networks Theorem 3.4.2 for the case of networks where some lower bounds are non- zero. 3.31. Show that a feasible circulation (if one exists) can always be found by just one max flow calculation in a suitable network. Hint: transform the network into an ( s, t )-flow network with all lower bounds equal to zero. 3.32. (+) Flows with balance vectors within prescribed intervals. Let N = ( V, A, l, u ) be a network where V = { 1 , 2 , . . . , n } and let a i b i , i = 1 , 2 . . . , n be integers. Prove that there exists a flow x in N which satisfies l ij x ij u ij ij A (3.40) a i b x ( i ) b i i V (3.41) if and only if the following three conditions are satisfied: X i V a i 0 (3.42) X i V b i 0 (3.43) u ( X, X ) l ( X, X ) + max { a ( X ) , - b ( X ) } ∀ X V, (3.44) where a ( X ) = i X a i . Hint: construct a network which has a feasible circulation if and only if (3.40) and (3.41) holds. Then apply Theorem 3.8.2.
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