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We consider the regular balanced model of formula generation in conjunctivenormal form (CNF) introduced by Boufkhad, Dubois, Interian, and Selman. We saythat 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 wedetermine upper bound on the threshold clause density such that there are no$p$-satisfying assignments with high probability above this upper bound. Thereare two aspects in deriving the lower bound using the second moment method. Thefirst aspect is, given any $p \in (0,1)$ and $k$, evaluate the lower bound onthe threshold. This evaluation is numerical in nature. The second aspect is toderive the lower bound as a function of $p$ for large enough $k$. We addressthe first aspect and evaluate the lower bound on the $p$-satisfying thresholdusing the second moment method. We observe that as $k$ increases the lowerbound seems to converge to the asymptotically derived lower bound for uniformmodel of formula generation by Achlioptas, Naor, and Peres.

Bounds on Thresholds Related to Maximum Satisfiability of Regular Random Formulas