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hwsol1.3

# hwsol1.3 - CSci 5403 Spring 2010 Brian Berzins Derived with...

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CSci 5403, Spring 2010 Brian Berzins Derived with: Stefan Nelson-Lindall, Hannah Jaber Homework 1.3 Solution (a) Show that quadeq is NP -complete. First: We show that quadeq NP by demonstrating that there is a polynomial time verifier for it. One such verifier takes the quadratic sys- tem and a variable assignment ~x ∈ { 0 , 1 } n and checks if each quadratic equation in the system evaluates to true. This requires m · n 2 (polyno- mial) time. Therefore quadeq NP . Second: We show that quadeq is NP -hard by a reduction from 3sat . Let φ be a 3sat boolean formula. Construct the reduction as follows: for each clause let the three literals be a , b , and c . Add the following equation to the system of quadratics: az 1 + bz 2 + cz 3 = 1 mod 2 where z 1 , z 2 and z 3 are variables that do not appear in any previous equation in our system. The variables a , b , and c represent the appro- priate literals in the clause and remain consistent across all clauses in φ . If a negated literal x appears, we substitute ( x +1) into this equation

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hwsol1.3 - CSci 5403 Spring 2010 Brian Berzins Derived with...

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