A New Bound for 3-Satisfiable MaxSat and its Algorithmic Application

TL;DR

This paper improves the lower bound on the number of satisfiable clauses in 3-satisfiable CNF formulas and introduces an efficient algorithmic approach with practical applications to a related parameterized problem.

Contribution

It presents a new bound for 3-satisfiable MaxSat formulas and develops a polynomial-time method to find a variable subset and assignment achieving this bound, with implications for fixed-parameter tractability.

Findings

Enhanced lower bound for clause satisfaction in 3-satisfiable formulas

Polynomial-time algorithm to identify key variable subset and assignment

Fixed-parameter tractability and linear kernel for 3-S-MAXSAT-AE

Abstract

Let F be a CNF formula with n variables and m clauses. F is 3-satisfiable if for any 3 clauses in F, there is a truth assignment which satisfies all of them. Lieberherr and Specker (1982) and, later, Yannakakis (1994) proved that in each 3-satisfiable CNF formula at least 2/3 of its clauses can be satisfied by a truth assignment. We improve this result by showing that every 3-satisfiable CNF formula F contains a subset of variables U, such that some truth assignment $\tau$ will satisfy at least $2m/3+ m_U/3+\rho n'$ clauses, where m is the number of clauses of F, m_U is the number of clauses of F containing a variable from U, n' is the total number of variables in clauses not containing a variable in U, and \rho is a positive absolute constant. Both U and $\tau$ can be found in polynomial time. We use our result to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. In 3-S-MAXSAT-AE, we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 2m/3 + k clauses, where k is the parameter.