We introduce $\mathsf{AmCMSO}$, an extension of counting monadic second-order logic ($\mathsf{CMSO}$) with predicates that refer to minimum- and maximum-value satisfying assignments. We establish fixed-parameter tractable model-checking meta-theorems for $\mathsf{AmCMSO}_1$ on graphs of bounded clique-width and for $\mathsf{AmCMSO}_2$ on graphs of bounded treewidth. These meta-theorems yield fixed-parameter tractable algorithms for several bilevel graph optimization problems, including interdiction and preassignment problems for solution uniquification, as well as algorithms for maximizing the diversity of optimal solutions without parameterizing by the optimum value. In contrast, allowing an optimality predicate to depend on an external set variable makes model checking hard for every level of the polynomial hierarchy, even on trees of fixed depth.
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