Maximum And- vs. Even-SAT

A multiset of literals, called a clause, is \emph{strongly satisfied} by an assignment if \emph{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 multiset of clauses that admits an assignment that strongly satisfies $ρ$ of the clauses, an assignment in which at least $ρ$ of the clauses are \emph{weakly satisfied}, in the sense that an \emph{even} number of literals evaluate to false. In particular, this implies an efficient algorithm for finding an undirected cut of value $ρ$ in a graph $G$ given that a directed cut of value $ρ$ in $G$ is promised to exist. A similar argument also gives an efficient algorithm for finding an acyclic subgraph of $G$ with $ρ$ edges under the same promise.

💡 Research Summary

The paper introduces a new promise optimization problem called Max‑And‑Even. An instance consists of a multiset of clauses, each clause being a multiset of literals (variables with a sign). The promise is that there exists an assignment of Boolean values (+1 for true, –1 for false) that strongly satisfies at least ρ clauses, meaning that every literal in each of those clauses evaluates to true. The algorithmic goal is to find an assignment that weakly satisfies at least the same number ρ of clauses, where a clause is weakly satisfied if an even number of its literals evaluate to false (equivalently, the product of the literals is +1).

The authors first express the objective as
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