Search versus Decision for $\mathsf{S}_2^\mathsf{P}$

TL;DR

This paper compares the complexity of search and decision problems in the class S2P, revealing a significant contrast and implications for the containment of certain complexity classes.

Contribution

It establishes that the search problem for S2P is equivalent to TFNP^NP, contrasting with the decision problem's known complexity, and explores implications for class containments.

Findings

Search problem for S2P is equivalent to TFNP^NP

Decision problem for S2P is in ZPP^NP

If search reduces to decision, then $oldsymbol{ ext{S}_2^ ext{P}}$ relates to $oldsymbol{ ext{ZPP}^ ext{NP}}$ and class containments.

Abstract

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 $\Sigma_2^p \cap \Pi_2^p$ is contained in ZPP^NP.