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Austrin showed that the approximation ratio $\beta\approx 0.94016567$obtained by the MAX 2-SAT approximation algorithm of Lewin, Livnat and Zwick(LLZ) is optimal modulo the Unique Games Conjecture (UGC) and modulo aSimplicity Conjecture that states that the worst performance of the algorithmis obtained on so called simple configurations. We prove Austrin's conjecture,thereby showing the optimality of the LLZ approximation algorithm, relying onlyon the Unique Games Conjecture. Our proof uses a combination of analytic andcomputational tools.  We also present new approximation algorithms for two restrictions of the MAX2-SAT problem. For MAX HORN-$\{1,2\}$-SAT, i.e., MAX CSP$(\{x\lor y,\bar{x}\lory,x,\bar{x}\})$, in which clauses are not allowed to contain two negatedliterals, we obtain an approximation ratio of $0.94615981$. For MAXCSP$(\{x\lor y,x,\bar{x}\})$, i.e., when 2-clauses are not allowed to containnegated literals, we obtain an approximation ratio of $0.95397990$. By adaptingAustrin's and our arguments for the MAX 2-SAT problem we show that these twoapproximation ratios are also tight, modulo only the UGC conjecture. Thiscompletes a full characterization of the approximability of the MAX 2-SATproblem and its restrictions.

Tight approximability of MAX 2-SAT and relatives, under UGC