Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 37 / 6 / c

Use the usual nonempty-clause convention and normalize repeated literals within a clause; tautologies are always satisfied. If empty clauses are admitted, discard them first: they contribute nothing to any assignment or to the optimum. Let $m$ be the resulting number of clause occurrences.

Assign independent fair truth values. A singleton clause is satisfied with probability $1/2$, a proper two-variable clause with probability $3/4$, and a tautology with probability one. By linearity of expectation, the expected number satisfied is at least $m/2$, hence at least $\mathrm{OPT}/2$.

To derandomize, use the method of conditional probabilities. After some variables have been fixed, let $F$ be the conditional expected number of satisfied clauses. For the next variable the two conditional expectations $F_0,F_1$ satisfy $F=(F_0+F_1)/2$. Fix the value with the larger expectation. This never decreases $F$. When all variables have been fixed, $F$ is the actual integer number of satisfied clauses, soCompute each conditional expectation by summing the probabilities of the clauses. Each has at most two variables, so its contribution is computed in constant time; scanning all clauses for each variable gives $O(nm)$ arithmetic operations. This is a polynomial-time approximation algorithm with approximation ratio $1/2$.