A Polynomial Time Delta-Decomposition Algorithm for Positive DNFs

TL;DR

This paper presents a polynomial-time algorithm for decomposing positive DNF formulas into components sharing a specified set of variables, with applications in game theory and boolean function minimization.

Contribution

It introduces a novel polynomial-time method for Delta-decomposition of positive DNFs using multilinear boolean polynomial factorization.

Findings

Finest Delta-decomposition components can be computed in polynomial time.

Provides an algorithm based on multilinear boolean polynomial factorization.

Enhances understanding of boolean function minimization and applications in game theory.

Abstract

We consider the problem of decomposing a positive DNF into a conjunction of DNFs, which may share a (possibly empty) given set of variables Delta. This problem has interesting connections with traditional applications of positive DNFs, e.g., in game theory, and with the broad topic of minimization of boolean functions. We show that the finest Delta-decomposition components of a positive DNF can be computed in polynomial time and provide a decomposition algorithm based on factorization of multilinear boolean polynomials.