Maximum And- vs. Even-SAT

TL;DR

This paper introduces an efficient algorithm for approximating solutions to complex satisfiability and graph cut problems, bridging the gap between strongly and weakly satisfied clauses and cuts.

Contribution

It presents a simple algorithm that guarantees at least as many weak satisfactions as the maximum strongly satisfied clauses, with applications to graph cuts and acyclic subgraphs.

Findings

Efficient approximation for maximum strongly satisfied clauses.

Algorithm guarantees at least as many weak satisfactions as maximum strong satisfactions.

Applicable to finding undirected cuts and acyclic subgraphs in graphs.

Abstract

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 $\rho$ of the clauses, an assignment in which at least $\rho$ 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 $\rho$ 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.