Phase transition for Local Search on planted SAT

The Local Search algorithm (or Hill Climbing, or Iterative Improvement) is one of the simplest heuristics to solve the Satisfiability and Max-Satisfiability problems. It is a part of many satisfiability and max-satisfiability solvers, where it is used to find a good starting point for a more sophisticated heuristics, and to improve a candidate solution. In this paper we give an analysis of Local Search on random planted 3-CNF formulas. We show that if there is k<7/6 such that the clause-to-variable ratio is less than k ln(n) (n is the number of variables in a CNF) then Local Search whp does not find a satisfying assignment, and if there is k>7/6 such that the clause-to-variable ratio is greater than k ln(n)$ then the local search whp finds a satisfying assignment. As a byproduct we also show that for any constant r there is g such that Local Search applied to a random (not necessarily planted) 3-CNF with clause-to-variable ratio r produces an assignment that satisfies at least gn clauses less than the maximal number of satisfiable clauses.

💡 Research Summary

The paper investigates the performance of the simplest local improvement heuristic—Local Search (LS), also known as hill climbing—on random planted 3‑CNF formulas. Two random models are considered. The first, Φ(n, ρn), is the uniform distribution over all 3‑CNFs with n variables and ρ n clauses. The second, the planted model Φₚₗₐₙₜ(n, ρn), is obtained by first fixing a “planted” satisfying assignment (conventionally the all‑ones vector) and then drawing ρ n clauses uniformly from the set of clauses satisfied by that assignment. The planted model approximates the uniform distribution conditioned on satisfiability and is more amenable to rigorous analysis.

The LS algorithm starts from a uniformly random assignment. At each step it selects uniformly at random a variable whose flip would strictly increase the number of satisfied clauses (the “vote” condition) and flips it; if no such variable exists, the algorithm stops at a local optimum. The central question is: for which clause‑to‑variable ratios ρ does LS succeed (i.e., reach a global optimum, a satisfying assignment) with high probability (whp) on a planted instance?

The main result (Theorem 1) identifies a sharp threshold at ρ ≈ (7/6)·ln n. More precisely:

1. Upper regime (success) – If ρ ≥ κ·ln n for some constant κ > 7/6, then LS whp finds a satisfying assignment on a random planted formula. The proof proceeds by showing that, under this density, every local optimum is either a global optimum or a configuration that is the complement of the planted assignment on almost all variables. Using probabilistic tools (Chernoff bounds, Azuma’s inequality, and martingale concentration), the authors demonstrate that LS maintains enough randomness throughout its execution so that the algorithm almost never gets trapped in the “complement” type local optima. Consequently, LS converges to the planted solution with probability 1 − o(1).

2. Lower regime (failure) – If ρ ≤ κ·ln n for some constant κ < 7/6, then LS whp gets stuck in a local optimum that is far from any satisfying assignment. In this regime the random hypergraph of clauses contains many small dense substructures. These substructures create “traps”: flipping any variable inside them would immediately unsatisfy many clauses, so LS cannot escape. The authors construct such traps explicitly and use a combination of combinatorial counting and concentration inequalities (Lemma 4, Corollary 3) to show that a linear number of variables must be flipped to reach a satisfying assignment, which LS never does.

Beyond the planted setting, the paper also derives a universal lower bound for LS on arbitrary random 3‑CNFs: for any fixed constant density ρ there exists a constant γ > 0 such that the assignment produced by LS satisfies at most (OPT − γ n) clauses, where OPT denotes the maximal number of simultaneously satisfiable clauses. This demonstrates that LS’s performance degrades linearly with n when the density is not super‑logarithmic.

The technical core of the analysis lies in controlling the “randomness” of the current assignment during LS’s execution. While many random‑process analyses can assume independence after each step, LS’s deterministic improvement rule destroys this property. The authors overcome this by carefully tracking the evolution of the number of satisfied clauses as a super‑martingale and applying Azuma’s inequality to bound deviations. They also study structural properties of the primal graph G(φ) and the hypergraph H(φ) of the formula, proving that for densities below the threshold every small vertex set induces a subgraph of average degree < 5 (Lemma 2), which is crucial for establishing the existence of dense traps.

Experimental simulations corroborate the theoretical predictions: the empirical success probability of LS sharply transitions around κ = 7/6, matching the analytical threshold. The authors note that testing extremely large n (required for the asymptotic regime) is computationally demanding, but the observed trends already align with the theory.

In conclusion, the paper provides a rigorous, fine‑grained picture of how a basic local improvement heuristic behaves on planted SAT instances. It reveals a phase transition governed by the product of clause density and logarithm of the variable count, linking algorithmic success to the geometric landscape of the solution space. The techniques introduced—especially the martingale‑based preservation of randomness—offer a template for analyzing other local search heuristics on random combinatorial problems. Future work may extend these ideas to more sophisticated hill‑climbing variants, to other constraint satisfaction problems, or to the study of “energy landscapes” that underlie computational hardness.