Extended formulations in combinatorial optimization

This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By “perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequ...

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Bibliographic Details
Published in:Annals of operations research Vol. 204; no. 1; pp. 97 - 143
Main Authors: Conforti, Michele, Cornuéjols, Gérard, Zambelli, Giacomo
Format: Journal Article
Language:English
Published: Boston Springer US 01.04.2013
Springer
Springer Nature B.V
Subjects:
Business and Management
Combinatorial analysis
Combinatorial optimization
Combinatorics
Discretization
Dynamic programming
Formulations
Inequalities
Inequalities (Mathematics)
Integer programming
Knapsack problem
Operations research
Operations Research/Decision Theory
Optimization
Polyhedrons
Studies
Theorems
Theory of Computation
Variables
United States
Italy
Combinatorial optimization polyhedral combinatorics
Extended formulations
ISSN:0254-5330, 1572-9338
Online Access:Get full text
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Summary:This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By “perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequalities that is exponential in the size of the data needed to describe the problem. Here we are particularly interested in situations where the addition of a polynomial number of extra variables allows a formulation with a polynomial number of inequalities. Such formulations are called “compact extended formulations”. We survey various tools for deriving and studying extended formulations, such as Fourier’s procedure for projection, Minkowski-Weyl’s theorem, Balas’ theorem for the union of polyhedra, Yannakakis’ theorem on the size of an extended formulation, dynamic programming, and variable discretization. For each tool that we introduce, we present one or several examples of how this tool is applied. In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set, and we present the proof of Fiorini, Massar, Pokutta, Tiwary and de Wolf of an exponential lower bound for the cut polytope. We also present Bienstock’s approximate compact extended formulation for the knapsack problem, Goemans’ result on the size of an extended formulation for the permutahedron, and the Faenza-Kaibel extended formulation for orbitopes.
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ISSN:0254-5330
1572-9338
DOI:10.1007/s10479-012-1269-0