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Chapter 7: Q6E (page 240)
Give an example of a linear program in two variables whose feasible region is infinite, but such that there is an optimum solution of bounded cost.
Example 1:
Example 2:
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Direct bipartite matching. We鈥檝e seen how to find a maximum matching in a bipartite graph via reduction to the maximum flow problem. We now develop a direct algorithm.
Let be a bipartite graph (so each edge has one endpoint in and one endpoint in ), and letbe a matching in the graph (that is, a set of edges that don鈥檛 touch). A vertex is said to be covered byif it is the endpoint of one of the edges in . An alternating path is a path of odd length that starts and ends with a non-covered vertex, and whose edges alternate between and .
(a) In the bipartite graph below, a matching is shown in bold. Find an alternating path.
(b) Prove that a matchingis maximal if and only if there does not exist an alternating path with respect to it.
(c) Design an algorithm that finds an alternating path intime using a variant of breadth-first search.
(d) Give a directalgorithm for finding a maximal matching in a bipartite graph.
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
Hall鈥檚 theorem. Returning to the matchmaking scenario of Section 7.3, suppose we have a bipartite graph with boys on the left and an equal number of girls on the right. Hall鈥檚 theorem says that there is a perfect matching if and only if the following condition holds: any subset of boys is connected to at least girls.
Prove this theorem. (Hint: The max-flow min-cut theorem should be helpful.)
There are many common variations of the maximum flow problem. Here are four of them.
(a) There are many sources and many sinks, and we wish to maximize the total flow from all sources to all sinks.
(b) Each vertex also has a capacity on the maximum flow that can enter it.
(c) Each edge has not only a capacity, but also a lower bound on the flow it must carry.
(d) The outgoing flow from each node u is not the same as the incoming flow, but is smaller by a factor of , whererole="math" localid="1659789093525" is a loss coefficient associated with node u.
Each of these can be solved efficiently. Show this by reducing (a) and (b) to the original max-flow problem, and reducing (c) and (d) to linear programming.
An edge of a flow network is called critical if decreasing the capacity of this edge results in a decrease in the maximum flow. Give an efficient algorithm that finds a critical edge in a network
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