Perfect Formulations

  • Michele Conforti
  • Gérard Cornuéjols
  • Giacomo Zambelli
Part of the Graduate Texts in Mathematics book series (GTM, volume 271)


A perfect formulation of a set \(S \subseteq \mathbb{R}^{n}\) is a linear system of inequalities Ax ≤ b such that \(\mathrm{conv}(S) =\{ x \in \mathbb{R}^{n}\,:\, Ax \leq b\}\). For example, Proposition 1.5 gives a perfect formulation of a 2-variable mixed integer linear set. When a perfect formulation is available for a mixed integer linear set, the corresponding integer program can be solved as a linear program. In this chapter, we present several classes of integer programming problems for which a perfect formulation is known. For pure integer linear sets, a classical case is when the constraint matrix is totally unimodular. Important combinatorial problems on directed or undirected graphs such as network flows and matchings in bipartite graphs have a totally unimodular constraint matrix.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Michele Conforti
    • 1
  • Gérard Cornuéjols
    • 2
  • Giacomo Zambelli
    • 3
  1. 1.Department of MathematicsUniversity of PadovaPadovaItaly
  2. 2.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA
  3. 3.Department of ManagementLondon School of Economics and Political ScienceLondonUK

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