Digraphs+Theory,+Algorithms+and+Applications_Part29 - 9.12...

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Unformatted text preview: 9.12 Exercises 543 9.59. Suppose D is a digraph which has k but not k +1 arc-disjoint out-branchings rooted at s and let F = { X ⊂ V- s : d- D ( X ) = k } . Explain how to find a minimal member of F (that is, no Y ⊂ X belongs to F ). Hint: first show how to find a member X of F using flows and then show how to find a minimal member inside X . For the later, see the result of Exercise 3.35 9.60. (+) Efficient implementation of the Frank-Fulkerson algorithm . Try to determine how efficient the Frank-Fulkerson algorithm can be im- plemented. I.e. identify places in the algorithm where a seemingly time con- suming step can be done efficiently. 9.61. Reducing the shortest path problem to the minimum cost branch- ing problem. Show how to reduce the shortest ( s,t )-path problem for di- graphs with non-negative weights on the edges to the min-cost out-branching problem. 9.62. Use your reduction from the previous problem to devise an algorithm for the shortest ( s,t )-path problem in a digraph with non-negative weights on the edges. That is, specialize the min-cost branching algorithm to the case where we only want to find a min-cost ( s,t )-path. Hint: how many minimal sets are there to choose from in each step of the algorithm? 9.63. Compare your algorithm above to Dijkstra’s algorithm (see Chapter 2) and other classical shortest path algorithms from Chapter 2. 9.64. Simplicity preserving augmentations for rooted arc-connectivity. Give an argument for the following claim. The Frank-Fulkerson algorithm can be used to find the cheapest set of new edges to add to a digraph to increase the maximum number of edge-disjoint out-branchings rooted at a fixed vertex from k to k + 1, even when we are not allowed to add arcs that are parallel to already existing ones. Hint: what intersecting family F and what digraph with the new possible arcs should we consider? 9.65. Increasing capacity of arcs to increase rooted arc-connectivity Prove that the Frank-Fulkerson algorithm also works if all new arcs have to be parallel to existing ones. 9.66. Show that if L is a laminar family (i.e. X,Y ∈ L implies X ∩ Y = ∅ , or X ⊂ Y , or Y ⊂ X ) on a ground set of size n , then the number of sets in L is at most 2 n- 1. Then show that indeed there are digraphs for which the Frank-Fulkerson algorithm may be run (legally) so that it will find 2 n- 1 sets before terminating phase 1. 9.67. Comparing the Frank-Fulkerson algorithm with classical minimum spanning tree algorithms. Suppose D = ( V,A ) is a symmetric digraph (i.e. xy ∈ E if and only if yx ∈ E ) and that c : A → R + satisfies c ( xy ) = c ( yx ). Compare the actions of the min-cost branching algorithm to well-known al- gorithms for finding a minimum spanning tree in a weighted (undirected) graph G . Such algorithms can be found in the book by Cormen, Leiserson and Rivest [169]....
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Digraphs+Theory,+Algorithms+and+Applications_Part29 - 9.12...

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