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Unformatted text preview: 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 Maxflow 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 FordFulkerson method. (b) Dinic’s algorithm. (c) The preflowpush algorithm. (d) The MKMalgorithm 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 realvalued flow . Let N = ( V,A,l,u ) be a network with source s and sink t and all data on the arcs nonnegative integers (note that some of the lower bounds may be nonzero). Suppose x is a realvalued feasible flow in N such that x ij is a noninteger for at least one arc. (a) Prove that there exists a feasible integer flow x in N with the property that  x ij x 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 ,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 nonzero lower bounds . Show how to modify the definition of x ⊕ ˜ x in order to obtain an analogue of 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 ≤ (3.42) X i ∈ V b i ≥ (3.43) u ( X, X ) ≥ l ( X,X ) + max { a ( X ) , b ( X ) } ∀ X ⊂ V, (3.44) where a ( X ) = ∑ i ∈ X a i ....
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This document was uploaded on 08/10/2011.
 Spring '11
 Algorithms

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