# Perfect Formulations

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

## 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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© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Michele Conforti
• 1
• Gérard Cornuéjols
• 2
• Giacomo Zambelli
• 3