The Projection Games Conjecture and the Hardness of Approximation of SSAT and related problems
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The Super-SAT or SSAT problem was introduced by Dinur, Kindler, Raz and Safra[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 factor n^c for some constant c>0, 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 the Projection Games Conjecture. We also reduce SSAT to the Nearest Codeword Problem (NCP) and Learning Halfspace Problem (LHP), as considered by Arora, Babai, Stern and Sweedyk[1997]. This proves that both these problems are NP-hard to approximate within factor N^c'/loglog n for some constant c'>0 where N is the size of the instances of the respective problems. Assuming the Projection Games Conjecture these problems are proved to be NP-hard to approximate within polynomial factors.
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