The backtracking survey propagation algorithm for solving random K-SAT problems

TL;DR

This paper reviews the backtracking survey propagation algorithm's effectiveness in solving large, hard random K-SAT problems near the phase transition, highlighting its ability to find unfrozen solutions efficiently.

Contribution

It demonstrates that the backtracking survey propagation algorithm can solve large K-SAT instances near the threshold in linear time, finding only unfrozen solutions and supporting conjectures about problem hardness.

Findings

The algorithm finds solutions close to the SAT-UNSAT threshold.

All solutions found have no frozen variables.

Linear-time solvability is linked to the absence of frozen variables.

Abstract

Discrete combinatorial optimization has a central role in many scientific disciplines, however, for hard problems we lack linear time algorithms that would allow us to solve very large instances. Moreover, it is still unclear what are the key features that make a discrete combinatorial optimization problem hard to solve. Here we study random K-satisfiability problems with $K=3,4$, which are known to be very hard close to the SAT-UNSAT threshold, where problems stop having solutions. We show that the backtracking survey propagation algorithm, in a time practically linear in the problem size, is able to find solutions very close to the threshold, in a region unreachable by any other algorithm. All solutions found have no frozen variables, thus supporting the conjecture that only unfrozen solutions can be found in linear time, and that a problem becomes impossible to solve in linear time when all solutions contain frozen variables.