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3SAT2VC - The following appears to be a little hand waving...

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The following appears to be a little hand waving. But it's not really. It is, though, necessary to sit down and draw some pictures and follow it through. Not all "proofs" can be simply "read." Some take work to follow. 3–SAT Vertex Cover (VC) Let I 3-SAT be an arbitrary instance of 3-SAT. For integers n and m, U = {u 1 , u 2 , …, u n } and C i = [z i1 , z i2 , z i3 } for 1 ≤ i ≤ m, where each z ij is either a u k or u k ' for some k. Construct an instance of VC as follows. For 1 ≤ i ≤ n construct 2n vertices, u i and u i ' with an edge between them. For each clause C i = [z i1 , z i2 , z i3 }, 1 ≤ i ≤ m, construct three vertices z i1 , z i2 , and z i3 and form a "triangle on them. Each z ij is one of the Boolean variables u k or it's complement u k '. Draw an edge between z ij and the Boolean variable (whichever it is) Each z ij has degree 3. Finally, set k = n+2m. Theorem. The given instance of 3-SAT is satisfiable if and only if the constructed instance of VC has a vertex cover with at most k vertices.
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