hwk5 - CS215 REVIEW EXERCISES not graded Problem 1(problem...

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Unformatted text preview: CS215 REVIEW EXERCISES not graded Problem 1. (problem 7.15) Let UNARY-SSUM be the subset-sum problem in which all numbers are represented in unary. That is, each number k is encoded as a string of k “1”s. (a) Show that UNARY-SSUM is in P. (b) Why does the NP-completeness proof for SUBSET-SUM in the book fail to show UNARY-SSUM is NP- complete? Be specific as you can about what is wrong with the content of the reduction. (Don’t just say, for example, “because UNARY-SSUM is in P, so if the reduction was correct, we would know that P=NP, and we don’t know that.”) Problem 2. (a) (problem 7.17) Prove, using the definitions of P, NP, and NP-complete, that if P=NP, then each language L in P (except the languages Σ * and ∅ ) is NP-complete. (b) Prove formally that ∅ is not NP-complete (according to the definition of NP-complete). Problem 3. (problem 7.24) Define D = {h p ( x 1 , x 2 , . . . , x n ) i : p is a polynomial over variables x 1 , x 2 , . . . , x n and p ( c 1 , c 2 , . . . , c n ) = 0 for some c 1 , c 2 , . . . , c n ∈ N } . That is, D contains the multi-variable polynomials that have integer roots. (a) Show that D is NP-hard. Hint: f ( x ) g ( x ) = 0 iff f ( x ) = 0 or g ( x ) = 0, while f ( x ) 2 + g ( x ) 2 = 0 iff f ( x ) = 0 and g ( x ) = 0....
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hwk5 - CS215 REVIEW EXERCISES not graded Problem 1(problem...

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