Finite-size scaling in random $K$-satisfiability problems

We provide a comprehensive view of various phase transitions in random $K$-satisfiability problems solved by stochastic-local-search algorithms. In particular, we focus on the finite-size scaling (FSS) exponent, which is mathematically important and practically useful in analyzing finite systems. Using the FSS theory of nonequilibrium absorbing phase transitions, we show that the density of unsatisfied clauses clearly indicates the transition from the solvable (absorbing) phase to the unsolvable (active) phase as varying the noise parameter and the density of constraints. Based on the solution clustering (percolation-type) argument, we conjecture two possible values of the FSS exponent, which are confirmed reasonably well in numerical simulations for $2\le K \le 3$.

💡 Research Summary

The paper presents a systematic study of phase transitions that occur when stochastic‑local‑search (SLS) algorithms are applied to random K‑satisfiability (K‑SAT) problems, with a particular focus on finite‑size scaling (FSS) exponents. The authors treat the random K‑SAT instance generation in the standard way, controlling two macroscopic parameters: the clause‑to‑variable ratio α (constraint density) and a noise parameter η that determines the probability of making a random variable flip at each step of the algorithm (the “temperature” of the search). By varying these two parameters, the system can be driven from a solvable (absorbing) phase, where the algorithm eventually reaches a configuration with zero unsatisfied clauses, to an unsolvable (active) phase, where the density of unsatisfied clauses remains finite indefinitely.

The central observable is the time‑averaged density of unsatisfied clauses, ρ(t). In the absorbing phase ρ(t) → 0, while in the active phase ρ(t) settles at a non‑zero value. The authors adopt the framework of nonequilibrium absorbing‑state phase transitions and write the standard finite‑size scaling ansatz for the order parameter and its susceptibility:

\