On the Complexity of Optimization Problems based on Compiled NNF Representations
This work addresses the problem of efficient optimization in AI and operations research by analyzing when off-line compilation of constraints reduces online complexity, but it is incremental as it builds on existing knowledge compilation frameworks.
The paper tackles the complexity of optimization problems where constraints are compiled into specific propositional languages (NNF subsets) and objective functions vary, identifying the computational complexity for each combination to determine when compilation eases finding optimal solutions.
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.