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

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.

Streamlining variational inference for constraint satisfaction problems

“…It has been shown that these instances have a very distinctive behavior where the probability of an instance having a solution has a phase transition explained as a function of the constraint density, α = m/n, for a problem with m clauses and n variables for large enough k. These instances exhibit a sharp crossover at a threshold density α s (k): they are almost always satisfiable below this threshold, and they become unsatisfiable for larger constraint densities [12,10]. Empirically, random instances with constraint density close to the satisfiability threshold are difficult to solve [23].…”

“…The base algorithm used in many state-of-the-art solvers for constraint satisfaction problems such as random k-SAT is survey inspired decimation [7,24,16,23]. The algorithm employs survey propagation, a message passing procedure that computes approximate single-variable marginal probabilities for use in a decimation procedure.…”

Theory of Nonequilibrium Local Search on Random Satisfaction Problems

“…We here take "work" to mean "find a solution in time scaling polynomially in system size", but keep it unspecified whether this has to happen always or with high probability, and leave aside rigorous considerations for which we refer to [5,6], and references cited therein. According to this criterion the best algorithm for random K-SAT is "survey propagation" [7] which in its most recent version is able to find solutions extremely close to the SAT/UNSAT threshold [8]. Survey propagation is however a quite complex algorithm tailored to random constraint satisfaction problems, and is not competitive on most real-world problems [9].…”

Relaxed Random Search for Solving K-Satisfiability and its Information Theoretic Interpretation

“…Backtracking to the earlier assignments and change them in the controlled way, is the common them of all these strategies [11]. DPLL has experienced many significant improvements over the years based on these backtracking techniques.…”

“…After this primary introduction about K-SAT, different strategies which have been designed to solve k-SAT are reviewed in Section 2. Section 3 focuses on randomized algorithm, in which it is tried to improve the time complexity of algorithm by relaxing the conditions imposed on random walking in the solution space inspired by the recent studies about the typical time complexity of K-SAT problem [11]. Finally in Section 4, concluding remarks and future works will be discussed.…”