Exercises 11 - CPSC 413 F18.pdf

# Set cover input a set u u 1 u 2 u n a collection of

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Set Cover Input: A set U = { u 1 , u 2 , . . . , u n } , a collection of subsets S 1 , S 2 , . . . , S m of U , and an integer 0 r m . Output: “Yes” if there exists a collection of at most r of the subsets S i whose union is U “No” otherwise. A set cover is a collection of subsets R 1 , R 2 , . . . , R so that j =1 R j = U . This decision problem is thus called Set Cover . 1. Show that Set Cover 6 m P SAT . 8. DNF formulas Problem 2 on DNF formulas on Final Examination for Winter 2007. Contrast this problem with 3SAT . Discuss why this problem does not imply that 3SAT can be solved in polynomial time. 9. Hitting Set Show that Set Cover 6 m P Hitting Set . Argue that your reduction f is correct, and argue briefly that your reduction runs in polynomial time. Hitting Set Input: A set A = { a 1 , a 2 , . . . , a n } , a collection B 1 , B 2 , . . . , B m of subsets of A , and an integer 0 k n . Output: “Yes” if there is a hitting set H A for B 1 , B 2 , . . . , B m of size at most k “No” otherwise. A hitting set is a subset H A such that H B j 6 = for all 1 j m . See also Exercise 5 on page 506 in Chapter 8.

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10. Resource Reservation See exercise 4 on page 506 in Chapter 8. First note that two processes P i , P j can both be active if and only if the sets of resources they require are disjoint. Then show that Resource Reservation 6 m P Independent Set .
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• Fall '13
• GeoffCruttwell
• Graph Theory, NP-complete problems

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