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Unformatted text preview: 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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