Fault Tolerant Boolean Satisfiability

TL;DR

This paper investigates the computational complexity of finding delta-models, robust satisfying assignments, across various classes of Boolean formulas, revealing polynomial and NP-complete cases.

Contribution

It extends previous work by analyzing the complexity of delta-models in classes with known polynomial satisfiability algorithms, identifying which classes allow efficient detection.

Findings

2-SAT and Affine-SAT have polynomial-time algorithms for delta-models.

Testing delta-models in Horn-SAT is NP-complete.

Complexity varies significantly across different Boolean formula classes.

Abstract

A delta-model is a satisfying assignment of a Boolean formula for which any small alteration, such as a single bit flip, can be repaired by flips to some small number of other bits, yielding a new satisfying assignment. These satisfying assignments represent robust solutions to optimization problems (e.g., scheduling) where it is possible to recover from unforeseen events (e.g., a resource becoming unavailable). The concept of delta-models was introduced by Ginsberg, Parkes and Roy (AAAI 1998), where it was proved that finding delta-models for general Boolean formulas is NP-complete. In this paper, we extend that result by studying the complexity of finding delta-models for classes of Boolean formulas which are known to have polynomial time satisfiability solvers. In particular, we examine 2-SAT, Horn-SAT, Affine-SAT, dual-Horn-SAT, 0-valid and 1-valid SAT. We see a wide variation in the complexity of finding delta-models, e.g., while 2-SAT and Affine-SAT have polynomial time tests for delta-models, testing whether a Horn-SAT formula has one is NP-complete.