practice midterm 2

practice midterm 2 - CME305 Sample Midterm II 1 Matchings...

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Unformatted text preview: CME305 Sample Midterm II 1. Matchings and Vertex Covers (a) Define what a matching in G is. (b) Define what a vertex cover of G is. (c) Let M be a maximum matching and C a minimum vertex cover. Show that |M | |C| 2|M |. 2. Traveling Salesman Problem Assume that deciding whether a graph has a Hamiltonian cycle is NPComplete. Prove that the Traveling Salesman Problem is NP-Hard. 3. Lecture Attendance Planning A group of students want to minimize their lecture attendance by sending only one of the group to each of the n lectures. We have the following constraints: Each of the n lectures should be covered. Lecture i starts at time ai and ends at time bi . It takes rij time to commute from lecture i to lecture j. Assume all times rij as well as the duration of the lectures are in minutes and integers. Minimize the number of students that will attend lectures i.e. develop a flow based algorithm to identify the minimum number of students needed to cover all n lectures. 1 ...
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