Graph-based local elimination algorithms in discrete optimization

The aim of this paper is to provide a review of structural decomposition methods in discrete optimization and to give a unified framework in the form of local elimination algorithms (LEA). This paper is organized as follows. Local elimination algorithms for discrete optimization (DO) problems (DOPs) with constraints are considered; a classification of dynamic programming computational procedures is given. We introduce Elimination Game and Elimination tree. Application of bucket elimination algorithm from constraint satisfaction (CS) to solving DOPs is done. We consider different local elimination schemes and related notions. Clustering that merges several variables into single meta-variable defines a promising approach to solve DOPs. This allows to create a quotient (condensed) graph and apply a local block elimination algorithm. In order to describe a block elimination process, we introduce Block Elimination Game. We discuss the connection of aforementioned local elimination algorithmic schemes and a way of transforming the directed acyclic graph (DAG) of computational LEA procedure to the tree decomposition.