# The Projection Games Conjecture and the Hardness of Approximation of super-SAT and related problems

## TL;DR

This paper proves the NP-hardness of approximating the Super-SAT problem and related problems within certain factors, assuming the Projection Games Conjecture, thereby establishing strong hardness results for several computational problems.

## Contribution

It proves the Super-SAT conjecture assuming the Projection Games Conjecture and reduces SSAT to other problems, establishing their NP-hardness of approximation.

## Key findings

- Proves Super-SAT NP-hardness of approximation assuming PGC.
- Reduces SSAT to NCP and LHP, showing their NP-hardness of approximation.
- Establishes polynomial factor hardness for SVP and CVP assuming PGC.

## Abstract

The Super-SAT or SSAT problem was introduced by Dinur et al.(2002,2003) to prove the NP-hardness of approximation of two popular lattice problems - Shortest Vector Problem(SVP) and Closest Vector Problem(CVP). They conjectured that SSAT is NP-hard to approximate to within a factor of $n^c$ ($c>0$ is constant), where $n$ is the size of the SSAT instance. In this paper we prove this conjecture assuming the Projection Games Conjecture(PGC), given by Moshkovitz (2012). This implies hardness of approximation of SVP and CVP within polynomial factors, assuming PGC. We also reduce SSAT to the Nearest Codeword Problem(NCP) and Learning Halfspace Problem(LHP), as considered by Arora et al.(1997). This proves that both these problems are NP-hard to approximate within a factor of $N^{c'/\log\log n}$($c'>0$ is constant) where $N$ is the size of the instances of the respective problems. Assuming PGC these problems are proved to be NP-hard to approximate within polynomial factors.