Optimization hardness as transient chaos in an analog approach to constraint satisfaction

TL;DR

This paper models k-SAT problems as a continuous-time dynamical system, revealing that beyond a certain constraint density, the system exhibits transient chaos and fractal basin boundaries, which correlate with increased problem hardness.

Contribution

It introduces a novel analog dynamical system approach to k-SAT, linking solution search difficulty to transient chaos and fractal basin boundaries, and demonstrates solution-finding in polynomial time despite complex fluctuations.

Findings

System always finds solutions for satisfiable formulas.

Transient chaos appears beyond a constraint density threshold.

Solution search occurs in polynomial continuous-time despite exponential energy fluctuations.

Abstract

Boolean satisfiability [1] (k-SAT) is one of the most studied optimization problems, as an efficient (that is, polynomial-time) solution to k-SAT (for $k\geq 3$) implies efficient solutions to a large number of hard optimization problems [2,3]. Here we propose a mapping of k-SAT into a deterministic continuous-time dynamical system with a unique correspondence between its attractors and the k-SAT solution clusters. We show that beyond a constraint density threshold, the analog trajectories become transiently chaotic [4-7], and the boundaries between the basins of attraction [8] of the solution clusters become fractal [7-9], signaling the appearance of optimization hardness [10]. Analytical arguments and simulations indicate that the system always finds solutions for satisfiable formulae even in the frozen regimes of random 3-SAT [11] and of locked occupation problems [12] (considered among the hardest algorithmic benchmarks); a property partly due to the system's hyperbolic [4,13] character. The system finds solutions in polynomial continuous-time, however, at the expense of exponential fluctuations in its energy function.