On the Complexity of Optimization Problems based on Compiled NNF
  Representations
  Optimization is a key task in a number of applications. When the set of
feasible solutions under consideration is of combinatorial nature and described
in an implicit way as a set of constraints, optimization is typically NP-hard.
Fortunately, in many problems, the set of feasible solutions does not often
change and is independent from the user's request. In such cases, compiling the
set of constraints describing the set of feasible solutions during an off-line
phase makes sense, if this compilation step renders computationally easier the
generation of a non-dominated, yet feasible solution matching the user's
requirements and preferences (which are only known at the on-line step). In
this article, we focus on propositional constraints. The subsets L of the NNF
language analyzed in Darwiche and Marquis' knowledge compilation map are
considered. A number of families F of representations of objective functions
over propositional variables, including linear pseudo-Boolean functions and
more sophisticated ones, are considered. For each language L and each family F,
the complexity of generating an optimal solution when the constraints are
compiled into L and optimality is to be considered w.r.t. a function from F is
identified.
