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7.34sispersolutionFall09 - CPSC 421 Homework 6 Solutions...

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CPSC 421 Fall 2009 Homework 6: Solutions 7.17 of Sipser: Suppose that P = NP , and let A P be some languge that is not or Σ * . Clearly, P = NP and A P implies that A NP . It remains to be shown that every B NP is polynomial time reducible to A . Consider arbitrary B NP , noting that B P . To test if w B , run the polynomial time algorithm N deciding B . If N answers yes, output constant w A ; otherwise, output constant w / / A . Strings w and w / necessarily exist because both A and Σ * - A are nonempty. This algorithm runs in polynomial time because N runs in polynomial time, and the length of w and w / are constant. Therefore, B is polynomial time reducible to A and A is NP-complete. 7.28 of Sipser: Given an instance S, C of the SET-SPLITTING problem, and a coloring partition R B = S , check that each C i C contains at least one element from R and S in polynomial time. Therefore, SET-SPLITTING is in NP.
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