Symmetric Rule-Based Achlioptas Processes for Random $k$-SAT
Arnab Chatterjee
TL;DR
The paper studies symmetric online Achlioptas processes for random $k$-SAT, introducing an assignment-symmetric MIDDLE-HEAVY rule and a Threshold-Symmetric Hybrid rule to shift the satisfiability threshold beyond the classical $2^k\log 2$ bound. Using a fixed $2$-SAT projection and a two-type Galton–Watson branching process certificate, it derives thresholds of the form $\alpha_{SYM}(k,\ell)=1/Q$ with $Q=p_1+2\sqrt{p_0p_2}$, where $p_0,p_1,p_2$ are clause-type frequencies. It identifies minimal choices $(k,\ell)=(4,5),(5,4),(\ge 6,3)$ that strictly exceed the baseline, and shows the Hybrid rule can match prior biased-rule performance while maintaining symmetry. These results deepen the understanding of Achlioptas processes in random CSPs and suggest avenues for sharper certificates and extensions to other two-SAT-like problems.
Abstract
Inspired by the "power-of-two-choices" model from random graphs, we investigate the possibility of limited choices of online clause choices that could shift the satisfiability threshold in random $k$-SAT.Here, we introduce an assignment symmetric, non-adaptive, topology-oblivious online rule called \emph{MIDDLE-HEAVY}, that prioritizes balanced sign profile clauses.Upon applying a biased $2$-SAT projection and a two-type branching process certificate, we derive closed-form expressions for the shifted thresholds $α_{\textbf{SYM}}(k,\ell)$ for this algorithm.We show that minimal choices $\ell=5$ for $k=4$, $\ell=4$ for $k=5$, and $\ell=3$ for $k\ge 6$ suffice to exceed the asymptotic first-moment upper bound $\sim 2^k \ln 2$ for random $k$-SAT.Moreover, to bridge the gap with biased assignment rules used in maximum of the previous works in this context, we propose a hybrid symmetric biased rule that achieves thresholds comparable to prior work while maintaining symmetry.Our results advance the understanding of Achlioptas processes in random CSPs beyond classical graph-theoretic settings.