Graph-based local elimination algorithms in discrete optimization

TL;DR

This paper reviews structural decomposition methods in discrete optimization, introduces a unified framework of local elimination algorithms, and discusses various schemes including bucket, block elimination, and their connections to graph decompositions.

Contribution

It provides a comprehensive review and unification of local elimination algorithms for discrete optimization, including new concepts like Elimination Game and Block Elimination Game.

Findings

Introduces Elimination Game and Elimination tree concepts.

Shows how bucket elimination applies from constraint satisfaction to optimization.

Connects local elimination schemes with graph decompositions.

Abstract

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.