Exercises 11 - CPSC 413 F18.pdf

Well connected vertex cover input a graph g v e on n

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Well-Connected Vertex Cover Input: a graph G = ( V, E ) on n = | V | vertices an integer k Output: “Yes” if G has a well-connected vertex cover of size (at most) k . “No” otherwise. 1. Show that Vertex Cover 6 m P Well-Connected Vertex Cover . 4. Recruitment problem See Exercise 3 on page 505 in Chapter 8 in which the problem Recruitment is being defined. Show that Vertex Cover 6 m P Recruitment .
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5. Reduction to 3SAT Consider the following decision problem SillyPrb , which is to determine if an array contains an integer that is bigger than t . Silly Prb Input: an array A = ( a 1 , . . . , a n ) of n integers. an integer t > 0. Output: “Yes” if there exists an index 1 i n such that A [ i ] > t “No” otherwise. 1. Show that Silly Prb 6 m P 3SAT . ( Hint: This is not obvious. Double-check your proof.) 2. Discuss the implications of this statement. 6. SAT and 3SAT Consider SAT , defined as 3SAT , except that there are no constraints on the number of literals in each clause. Each clause contains one or more distinct literals. Show that SAT 6 m P 3SAT . ( Hint: Generalize the reduction of 4SAT 6 m P 3SAT .) 7. Set Cover Let U = { u 1 , u 2 , . . . , u n } be a set. Consider we are given a collection of subsets S 1 , S 2 , . . . , S m of U . That is, S i U for each 1 i m of the m subsets. We want to choose some of those m subsets so that the union of the chosen subsets is U .
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  • Fall '13
  • GeoffCruttwell
  • Graph Theory, NP-complete problems

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