Optimization and probabilistic satisfiability on nested and co‐nested formulas

Nested and co‐nested formulas are two classes of CNF instances that can be characterized in terms of outerplanar graphs. For these classes, linear time algorithms are known for SAT and (unweighted) Max‐SAT. In this paper we devise linear time algorithms for a general optimization version of SAT. Moreover, we show that a general probabilistic version of SAT reduces to solving a system of linear inequalities where the number of variables and constraints is linear in the size of the formula.