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Chapter 7: Q19E (page 242)
Suppose someone presents you with a solution to the max-flow problem on some network. Give a linear-time algorithm to determine whether the solution does indeed give a maximum flow.
Ford-Fulkerson algorithm is the linear time algorithm that determines if the solution obtains a maximum flow.
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Question: Consider the following simple network with edge capacities as shown.
a) Show that, if the Ford-Fulkerson algorithm is run on this graph, a careless choice of updates might cause it to take 1000iterations. Imagine if the capacities were a million instead of 1000.
We will now find a strategy for choosing paths under which the algorithm is guaranteed to terminate in a reasonable number of iterations.
Consider an arbitrary directed network in which we want to find the maximum flow.Assume for simplicity that all edge capacities are at least 1, and define the capacity of an s - t path to be the smallest capacity of its constituent edges. The fattest path from s to t is the path with the most capacity.
b) Show that the fattest s - t path in a graph can be computed by a variant of Dijkstra鈥檚 algorithm.
c) Show that the maximum flow in Gis the sum of individual flows along at mostpaths from s to t.
d) Now show that if we always increase flow along the fattest path in the residual graph, then the Ford-Fulkerson algorithm will terminate in at most iterations, where F is the size of the maximum flow. (Hint: It might help to recall the proof for the greedy set cover algorithm in Section 5.4.)
In fact, an even simpler rule鈥攆inding a path in the residual graph using breadth-first search鈥� guarantees that atmost iterations will be needed.
Consider the following network (the numbers are edge capacities).
(a)Find the maximum flow and a minimum cut.
(b)Draw the residual graph (along with its edge capacities). In this residual network, mark the vertices reachable from and the vertices from which is reachable.
(c)An edge of a network is called a bottleneck edge if increasing its capacity results in an increase in the maximum flow. List all bottleneck edges in the above network.
(d)Give a very simple example (containing at most four nodes) of a network which has no bottleneck edges.
(e)Give an efficient algorithm to identify all bottleneck edges in a network.
For the following network, with edge capacities as shown, find the maximum flow from S to T, along with a matching cut.
Question: A linear program for shortest path. Suppose we want to compute the shortest path from node s to node t in a directed graph with edge lengths .
a) Show that this is equivalent to finding an s - tflow fthat minimizes subject to size (f) = 1. There are no capacity constraints.
b) Write the shortest path problem as a linear program.
c) Show that the dual LP can be written as
role="math" localid="1659250472483"
d) An interpretation for the dual is given in the box on page 223. Why isn鈥檛 our dual LP identical to the one on that page?
Question: Duckwheat is produced in Kansas and Mexico and consumed in New York and California. Kansas produces 15 shnupells of duckwheat and Mexico 8. Meanwhile, New York consumes 10 shnupells and California 13. The transportation costs per shnupell are \(4 from Mexico to New York, \)1 from Mexico to California, \(2 from Kansas to New York, and \)3 and from Kansas to California. Write a linear program that decides the amounts of duckwheat (in shnupells and fractions of a shnupell) to be transported from each producer to each consumer, so as to minimize the overall transportation cost
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