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Optimization is a key task in a number of applications. When the set offeasible solutions under consideration is of combinatorial nature and describedin an implicit way as a set of constraints, optimization is typically NP-hard.Fortunately, in many problems, the set of feasible solutions does not oftenchange and is independent from the user's request. In such cases, compiling theset of constraints describing the set of feasible solutions during an off-linephase makes sense, if this compilation step renders computationally easier thegeneration of a non-dominated, yet feasible solution matching the user'srequirements and preferences (which are only known at the on-line step). Inthis article, we focus on propositional constraints. The subsets L of the NNFlanguage analyzed in Darwiche and Marquis' knowledge compilation map areconsidered. A number of families F of representations of objective functionsover propositional variables, including linear pseudo-Boolean functions andmore sophisticated ones, are considered. For each language L and each family F,the complexity of generating an optimal solution when the constraints arecompiled into L and optimality is to be considered w.r.t. a function from F isidentified.

On the Complexity of Optimization Problems based on Compiled NNF Representations