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Perfect Formulations

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Integer Programming

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 271))

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Abstract

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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Authors and Affiliations

  1. Department of Mathematics, University of Padova, Padova, Italy

    Michele Conforti

  2. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA

    Gérard Cornuéjols

  3. Department of Management, London School of Economics and Political Science, London, UK

    Giacomo Zambelli

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  1. Michele Conforti
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  2. Gérard Cornuéjols
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  3. Giacomo Zambelli
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Conforti, M., Cornuéjols, G., Zambelli, G. (2014). Perfect Formulations. In: Integer Programming. Graduate Texts in Mathematics, vol 271. Springer, Cham. https://doi.org/10.1007/978-3-319-11008-0_4

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