  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 a
Simplicity Conjecture that states that the worst performance of the algorithm
is obtained on so called simple configurations. We prove Austrin's conjecture,
thereby showing the optimality of the LLZ approximation algorithm, relying only
on the Unique Games Conjecture. Our proof uses a combination of analytic and
computational tools.
  We also present new approximation algorithms for two restrictions of the MAX
2-SAT problem. For MAX HORN-$\{1,2\}$-SAT, i.e., MAX CSP$(\{x\lor y,\bar{x}\lor
y,x,\bar{x}\})$, in which clauses are not allowed to contain two negated
literals, we obtain an approximation ratio of $0.94615981$. For MAX
CSP$(\{x\lor y,x,\bar{x}\})$, i.e., when 2-clauses are not allowed to contain
negated literals, we obtain an approximation ratio of $0.95397990$. By adapting
Austrin's and our arguments for the MAX 2-SAT problem we show that these two
approximation ratios are also tight, modulo only the UGC conjecture. This
completes a full characterization of the approximability of the MAX 2-SAT
problem and its restrictions.
