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On strong integrality properties of the perfect matching polytope

Author

Listed:
  • Grappe, Roland

    (Université Paris Dauphine)

  • Lacroix, Mathieu

    (Université Sorbonne Paris Nord)

  • Pisanu, Francesco

    (Université catholique de Louvain, LIDAM/CORE, Belgium)

Abstract

This paper investigates integrality properties of perfect matching polytopes, focusing on box-total dual integrality and integer decomposition properties. We begin by characterizing the graphs whose perfect matching polytope is a slice of the nonnegative orthant, identifying these as the solid graphs introduced by de Carvalho, Lucchesi, and Murty in “On a Conjecture of Lovász Concerning Bricks: I. The Characteristic of a Matching Covered Graph” (Journal of Combinatorial Theory, Series B). As a result, we show that the perfect matching polytope of solid graphs admits a compact description, and we establish that deciding the box-total dual integrality of a perfect matching polytope can be done in polynomial time. Additionally, we characterize the conditions under which perfect matching polytopes of two fundamentalgraphclasses,namelynear-bricksandbicriticalgraphs,arebox-totallydualintegral. We discuss implications of these results for identifying perfect matching polytopes with the integer decomposition property. This in particular unveils a new positive case of the generalized Berge-Fulkeron conjecture.

Suggested Citation

  • Grappe, Roland & Lacroix, Mathieu & Pisanu, Francesco, 2024. "On strong integrality properties of the perfect matching polytope," LIDAM Discussion Papers CORE 2024032, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2024032
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