Bounds on Thresholds Related to Maximum Satisfiability of Regular Random Formulas

TL;DR

This paper investigates thresholds for p-satisfiability in regular random CNF formulas, providing bounds using probabilistic methods and analyzing their behavior as clause size increases.

Contribution

It evaluates the p-satisfying threshold bounds for regular random formulas using the second moment method and compares them to known asymptotic bounds.

Findings

Upper bounds on p-satisfying thresholds established.

Lower bounds evaluated numerically for various p and k.

Bounds converge to known asymptotic bounds as k increases.

Abstract

We consider the regular balanced model of formula generation in conjunctive normal form (CNF) introduced by Boufkhad, Dubois, Interian, and Selman. We say that a formula is $p$-satisfying if there is a truth assignment satisfying $1-2^{-k}+p 2^{-k}$ fraction of clauses. Using the first moment method we determine upper bound on the threshold clause density such that there are no $p$-satisfying assignments with high probability above this upper bound. There are two aspects in deriving the lower bound using the second moment method. The first aspect is, given any $p \in (0,1)$ and $k$, evaluate the lower bound on the threshold. This evaluation is numerical in nature. The second aspect is to derive the lower bound as a function of $p$ for large enough $k$. We address the first aspect and evaluate the lower bound on the $p$-satisfying threshold using the second moment method. We observe that as $k$ increases the lower bound seems to converge to the asymptotically derived lower bound for uniform model of formula generation by Achlioptas, Naor, and Peres.