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Extended formulations in combinatorial optimization

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Abstract

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.

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Authors and Affiliations

  1. Università di Padova, Padua, Italy

    Michele Conforti & Giacomo Zambelli

  2. Carnegie Mellon University, Pittsburgh, PA, USA

    Gérard Cornuéjols

  3. Université d’Aix-Marseille, Marseille, France

    Gérard Cornuéjols

Authors
  1. Michele Conforti
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  2. Gérard Cornuéjols
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  3. Giacomo Zambelli
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Correspondence to Michele Conforti.

Additional information

Michele Conforti was supported by the Progetto di Eccellenza 2008–2009 of the Fondazione Cassa di Risparmio di Padova e Rovigo. Gérard Cornuéjols was supported by NSF grant CMMI0653419, ONR grant N00014-09-1-0133 and ANR grant BLAN06-1-138894. Giacomo Zambelli was supported by the Progetto di Eccellenza 2008–2009 of the Fondazione Cassa di Risparmio di Padova e Rovigo.

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Conforti, M., Cornuéjols, G. & Zambelli, G. Extended formulations in combinatorial optimization. 4OR-Q J Oper Res 8, 1–48 (2010). https://doi.org/10.1007/s10288-010-0122-z

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