Search versus Decision for $\mathsf{S}_2^\mathsf{P}$
By: Lance Fortnow
Potential Business Impact:
Finds harder computer problems than before.
We compare the complexity of the search and decision problems for the complexity class S2P. While Cai (2007) showed that the decision problem is contained in ZPP^NP, we show that the search problem is equivalent to TFNP^NP, the class of total search problems verifiable in polynomial time with an NP oracle. This highlights a significant contrast: if search reduces to decision for S2P, then $Σ_2^p \cap Π_2^p$ is contained in ZPP^NP.
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