Maximum And- vs. Even-SAT

A (multi)set of literals, called a clause, is strongly satisfied by an assignment if no literal evaluates to false. Finding an assignment that maximises the number of strongly satisfied clauses is NP-hard. We present a simple algorithm that finds, given a set of clauses that admits an assignment that strongly satisfies a $\rho$-fraction of the clauses, an assignment in which at least a $\rho$-fraction of the clauses is weakly satisfied, in the sense that an even number of literals evaluates to false. In particular, this implies an efficient algorithm for finding an undirected cut of value $\rho$ in a graph $G$ given that a directed cut of value $\rho$ in $G$ is promised to exist. A similar argument also gives an efficient algorithm for finding an acyclic subgraph of $G$ with $\rho$ edges under the same promise.