Local Backbones

TL;DR

This paper introduces local variants of backbones in CNF formulas, analyzes their computational complexity across different classes, and presents empirical evidence that structured SAT instances have many local backbones.

Contribution

It defines and studies the complexity of k-backbones and iterative k-backbones, providing new insights into local backbones in SAT problems.

Findings

Structured SAT instances have many local backbones.

Random SAT instances have few local backbones.

Complexity results vary across formula classes.

Abstract

A backbone of a propositional CNF formula is a variable whose truth value is the same in every truth assignment that satisfies the formula. The notion of backbones for CNF formulas has been studied in various contexts. In this paper, we introduce local variants of backbones, and study the computational complexity of detecting them. In particular, we consider k-backbones, which are backbones for sub-formulas consisting of at most k clauses, and iterative k-backbones, which are backbones that result after repeated instantiations of k-backbones. We determine the parameterized complexity of deciding whether a variable is a k-backbone or an iterative k-backbone for various restricted formula classes, including Horn, definite Horn, and Krom. We also present some first empirical results regarding backbones for CNF-Satisfiability (SAT). The empirical results we obtain show that a large fraction of the backbones of structured SAT instances are local, in contrast to random instances, which appear to have few local backbones.