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Extended formulations in combinatorial optimization

Conforti, Michele, Cornuéjols, Gérard and Zambelli, Giacomo (2010) Extended formulations in combinatorial optimization. 4or: a Quarterly Journal of Operations Research, 8 (1). pp. 1-48. ISSN 1619-4500

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Identification Number: 10.1007/s10288-010-0122-z


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. 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.

Item Type: Article
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Additional Information: © 2010 Springer-Verlag
Divisions: Management
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited: 24 Jan 2011 17:24
Last Modified: 18 Jun 2024 02:15

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