introduction-lp-duality1

1 model this problem as a graph problem 2 write a

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Unformatted text preview: ). A vertex cover is a set C of vertices such that all edges e of E are incident to at least one vertex of C. The Minimum vertex cover problem is to find a cover C of minimum size. 1. Express the minimum vertex cover problem for the following graph as a linear program: F E B C D A 2. Express the minimum vertex cover problem for a general graph as a linear program. Exercise 9.23 (Minimum Edge Cover). Adapt the linear program giving the minimum vertex cover to the minimum edge vertex cover problem. 154 CHAPTER 9. LINEAR PROGRAMMING Exercise 9.24 (Matching for Bipartite Graphs). Adapt the linear program giving the matching of maximum cardinality of a general graph to a bipartite graph. Exercise 9.25. Considering the graph A | B–D |/ C what does the following linear program do? min subject to xA + xB + xC + xD xA + xB ≥ 1 xB + xD ≥ 1 xB + xC ≥ 1 xC + xD ≥ 1 xA ≥ 0, xB ≥ 0, xC ≥ 0, xD ≥ 0 Exercise 9.26 (Maximum cardinality matching problem (Polynomial < flows or augmenting paths)). Let G = (V, E ) be a graph. A matching M ⊆ E is a collection of edges such that every vertex of V is incident to at most one edge of M . The maximum cardinality matching problem is to find a matching M of maximum size. Express the maximum cardinality matching problem as a linear program. Exercise 9.27 (Maximum clique (NP-complete)). A clique of a graph G = (V, E) is a subset C of V, such that every two nodes in V are joined by an edge of E. The maximum clique problem consist of finding the largest cardinality of a clique. Express the maximum clique problem as a linear program. Hint: think of a constraint for nodes not linked by an edge. Exercise 9.28 (Modeling). A university class has to go from Marseille to Paris using buses. There are some strong inimities inside the group and two people that dislike each other cannot share the same bus. What is the minimum number of buses needed to transport the whole group? Write a LP that solve the problem. (We suppose that a bus does not have a limitation on the number of places. ) Exercise 9.29 (French newspaper enigma). What is the maximum size of a set of integers between 1 and 100 such that for any pair (a,b), the difference a-b is not a square ? 1. Model this problem as a graph problem. 2. Write a linear program to solve it. 9.4. EXERCICES 155 Exercise 9.30 (Maximum independent set (NP-hard)). An independent set of a graph G = (V, E) is a subset I of V , such that every two nodes in V are not joined by an edge of E. The maximum independent set problem consist of finding the largest cardinality of an independent set. Exercise 9.31 (Minimum Set Cover (NP-hard)). Input: A universal set U = {1, ..., n} and a family S of subsets S1 , . . . , Sm of U . Optimization Problem: What is the smallest subset of subsets T ⊂ S such that ∪ti ∈T ti = U ? Decision problem: Given an integer k, does there exists a subset of T of cardinality k, such that ∪ti ∈T ti = U ? This decision problem is NP-complete. Question: Write the set cover problem as a linear program....
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