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5.1. In the Path Edge Cover problem, we are given a directed acyclic graph A with two distinguished nodes s and t. We wish to find a minimum number of directed s t paths that cover all edges, that is, every edge must be in at least one of the selected paths. (If no solution exists at all, this shall be recognized, too.) Clearly, this is a special case of the Set Cover problem. Show that Path Edge Cover is also a special case of our minimum flow problem fom Exercise 3. More precisely: Turn A in polynomial time into a graph G with appropriate lower and upper edge capacities, and briefly show equivalence of the problems. Do not forget that an equivalence has two directions.

5.1. In the Path Edge Cover problem, we are given a directed acyclic graph A with two distinguished nodes s and t. We wish to find a minimum number of directed s t paths that cover all edges, that is, every edge must be in at least one of the selected paths. (If no solution exists at all, this shall be recognized, too.) Clearly, this is a special case of the Set Cover problem. Show that Path Edge Cover is also a special case of our minimum flow problem fom Exercise 3. More precisely: Turn A in polynomial time into a graph G with appropriate lower and upper edge capacities, and briefly show equivalence of the problems. Do not forget that an equivalence has two directions.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 80EQ
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Exercise 5. 5.1. In the Path Edge Cover problem, we are given a directed acyclic graph A with two distinguished nodes s and t. We wish to find a minimum number of directed s- - t paths that cover all edges, that is, every edge must be in at least one of the selected paths. (If no solution exists at all, this shall be recognized, too.) Clearly, this is a special case of the Set Cover problem. Show that Path Edge Cover is also a special case of our minimum flow problem fom Exercise 3. More precisely: Turn A in polynomial time into a graph G with appropriate lower and upper edge capacities, and briefly show equivalence of the problems. Do not forget that an equivalence has two directions. 5.2.. Combine the previous results, to show that Path Edge Cover is solvable in polynomial time. In particular, state and motivate some polynomial time bound, as a function of the graph size. The time bound might depend on your approach to Exercise 4, but in any case it must be polynomial.
Transcribed Image Text:Exercise 5. 5.1. In the Path Edge Cover problem, we are given a directed acyclic graph A with two distinguished nodes s and t. We wish to find a minimum number of directed s- - t paths that cover all edges, that is, every edge must be in at least one of the selected paths. (If no solution exists at all, this shall be recognized, too.) Clearly, this is a special case of the Set Cover problem. Show that Path Edge Cover is also a special case of our minimum flow problem fom Exercise 3. More precisely: Turn A in polynomial time into a graph G with appropriate lower and upper edge capacities, and briefly show equivalence of the problems. Do not forget that an equivalence has two directions. 5.2.. Combine the previous results, to show that Path Edge Cover is solvable in polynomial time. In particular, state and motivate some polynomial time bound, as a function of the graph size. The time bound might depend on your approach to Exercise 4, but in any case it must be polynomial.
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