SAT Backdoors: Depth Beats Size

TL;DR

This paper introduces fixed-parameter tractable approximation algorithms for computing backdoor depth into Horn and Krom classes, enabling efficient satisfiability testing for formulas with bounded backdoor depth, which captures new tractable classes.

Contribution

It presents novel FPT approximation algorithms for backdoor depth, extending the understanding of tractable CNF formulas beyond size-based measures.

Findings

Algorithms for backdoor depth approximation into Horn and Krom.

Linear-time satisfiability decision for formulas with bounded backdoor depth.

Bounded backdoor depth captures new tractable classes of CNF formulas.

Abstract

For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by M\"{a}hlmann, Siebertz, and Vigny (2021), is a more refined distance measure, which admits the utilization of different backdoor variables in parallel. Bounded backdoor size implies bounded backdoor depth, but there are formulas of constant backdoor depth and arbitrarily large backdoor size. We propose FPT approximation algorithms to compute backdoor depth into the classes Horn and Krom. This leads to a linear-time algorithm for deciding the satisfiability of formulas of bounded backdoor depth into these classes. We base our FPT approximation algorithm on a sophisticated notion of obstructions, extending M\"{a}hlmann et al.'s obstruction trees in various ways, including the addition of separator obstructions. We develop the algorithm through a new game-theoretic framework that simplifies the reasoning about backdoors. Finally, we show that bounded backdoor depth captures tractable classes of CNF formulas not captured by any known method.