Firstly, we define A and B as two decision problems with the same set of inputs.
Define a new decision problem "A AND B" as follows: The input to "A AND B" is any valid input x for A and for B. The output is to decide if A(x) = Yes and B(x) = Yes. For example, in the "Independent Set AND Vertex Cover" problem, we must decide if G has an independent set of size >= k and a vertex cover of size <= k.
Find two decision problems A and B with the same set of inputs such that both A and B are NP-complete, but "A AND B" is computable in polynomial time.
My first try is SAT and UNSAT - thinking of two decision problems that cancel themselves out since if one is true, the other automatically isn't. Thus, the conjunction would always be false. I know that the NP-complete problems are not closed under union/intersection I am just struggling to find examples. I am wondering if this example works/does not work and if there are any other specific examples that would work as A and B.
