Order-to-chaos transition in the hardness of random Boolean satisfiability problems

Abstract

Transient chaos is a ubiquitous phenomenon characterizing the dynamics of phase-space trajectories evolving towards a steady-state attractor in physical systems as diverse as fluids, chemical reactions, and condensed matter systems. Here we show that transient chaos also appears in the dynamics of certain efficient algorithms searching for solutions of constraint satisfaction problems that include scheduling, circuit design, routing, database problems, and even Sudoku. In particular, we present a study of the emergence of hardness in Boolean satisfiability (k-SAT), a canonical class of constraint satisfaction problems, by using an analog deterministic algorithm based on a system of ordinary differential equations. Problem hardness is defined through the escape rate κ, an invariant measure of transient chaos of the dynamical system corresponding to the analog algorithm, and it expresses the rate at which the trajectory approaches a solution. We show that for a given density of constraints and fixed number of Boolean variables N, the hardness of formulas in random k-SAT ensembles has a wide variation, approximable by a lognormal distribution. We also show that when increasing the density of constraints α, hardness appears through a second-order phase transition at α_{χ} in the random 3-SAT ensemble where dynamical trajectories become transiently chaotic. A similar behavior is found in 4-SAT as well, however, such a transition does not occur for 2-SAT. This behavior also implies a novel type of transient chaos in which the escape rate has an exponential-algebraic dependence on the critical parameter κ∼N^{B|α-α_{χ}|^{1-γ}} with 0<γ<1. We demonstrate that the transition is generated by the appearance of metastable basins in the solution space as the density of constraints α is increased.

A continuous-time MaxSAT solver with high analog performance

“…This is repeated until all clauses are satisfied, for solvable SAT problems. The properties and performance of this solver have been discussed in previous publications 42 – 44 . In 43 we show that the notion of escape rate κ can be used to characterize the hardness of individual problem instances.…”

“…It was designed such that all the SAT solutions appear as attractive fixed points for the dynamics while no other attractors exist trapping the dynamics. For hard problems its behavior becomes chaotic, showing that problem hardness and chaos 43 , 44 are related notions within this context, and thus chaos theory can be used to study computational complexity. For fully satisfiable SAT problems the chaos is necessarily transient 45 , with the trajectory eventually settling onto one of its attracting fixed points (a SAT solution).…”

Dynamical Systems Theory and Algorithms for NP-hard Problems

“…In particular, we will cover the use of dynamical systems theory for constructing decentralized graph clustering algorithms [27,28], solutions for the TSP [29], and quantum-inspired networks of Duffing oscillators for solving the MAX-CUT problem [30]. We then switch to the use of dynamical systems theory for analysis of algorithms [31] and the underlying problems [32,33].…”

Phase transitions of the typical algorithmic complexity of the random satisfiability problem studied with linear programming

“…While this SAT-UNSAT transition is certainly the most scrutinized in the K -SAT problem, there exist more transitions. For example 3-SAT, where SAT-UNSAT occurs at α s ≈ 4.26 [9], shows a transition to chaotic behavior at α χ ≈ 3.28 [19], i.e., using a continuous time deterministic solver [20] the trajectory will find the solution if one exists, but it will show chaotic transient behavior above this threshold resulting in increasing escape rates from attractors. This leads to a higher computational cost and can therefore be used as a measure of hardness.…”

“…Thus LP somehow approaches for minimization problems the true feasible and optimum solution from below (In this sense the analog solver of Ref. [19, 20] also operates outside the feasible region). Nevertheless, a key observation is that whenever LP gives a feasible solution, it must be the true optimum solution of the combinatorial problem.…”