Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lance Fortnow

TL;DR

The paper investigates the contrast between search and decision for the complexity class $ ext{S}_2^ ext{P}$. It shows that while the decision problem for $ ext{S}_2^ ext{P}$ lies in $ ext{ZPP}^{ ext{NP}}$ (as previously known), the corresponding search problem is complete for $ ext{TFNP}^{ ext{NP}}$, with a bidirectional reduction between $ ext{TFNP}^{ ext{NP}}$ and $ ext{S}_2^ ext{P}$-Search. This establishes a deep equivalence: solving $ ext{S}_2^ ext{P}$-Search captures the full power of total search problems verifiable with an NP oracle, whereas the decision version remains comparatively easier. The work further implies that if search reduced to decision for $ ext{S}_2^ ext{P}$, then $ ext{Sigma}_2^ ext{P} igcap ext{Pi}_2^ ext{P} subseteq ext{ZPP}^{ ext{NP}}$ would hold unless unlikely collapses occur. Overall, $ ext{S}_2^ ext{P}$ emerges as the natural class for $ ext{TFNP}^{ ext{NP}}$-level total search problems with NP-verifiable witnesses.

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 $Σ_2^p \cap Π_2^p$ is contained in ZPP^NP.

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

TL;DR

The paper investigates the contrast between search and decision for the complexity class . It shows that while the decision problem for lies in (as previously known), the corresponding search problem is complete for , with a bidirectional reduction between and -Search. This establishes a deep equivalence: solving -Search captures the full power of total search problems verifiable with an NP oracle, whereas the decision version remains comparatively easier. The work further implies that if search reduced to decision for , then would hold unless unlikely collapses occur. Overall, emerges as the natural class for -level total search problems with NP-verifiable witnesses.

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 is contained in ZPP^NP.

Paper Structure

This paper contains 8 sections, 2 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. The Equivalence
  5. $\mathsf{S}_2^\mathsf{P}$-Search in $\mathsf{TFNP}^{\mathsf{NP}}$
  6. $\mathsf{TFNP}^{\mathsf{NP}}$ reduces to $\mathsf{S}_2^\mathsf{P}$-Search
  7. Implications and Discussion
  8. Conclusion

Key Result

Theorem 4

$\mathsf{S}_2^\mathsf{P}$-Search is equivalent to $\mathsf{TFNP}^{\mathsf{NP}}$.

Theorems & Definitions (6)

  • Definition 1: $\mathsf{S}_2^\mathsf{P}$
  • Definition 2: $\mathsf{S}_2^\mathsf{P}$-Search
  • Definition 3: $\mathsf{TFNP}^{\mathsf{NP}}$
  • Theorem 4
  • Corollary 5
  • proof