Easy and hard separation of sparse and dense odd-set constraints in matching

We investigate polytopes intermediate between the fractional matching and the perfect matching polytopes, by imposing a strict subset of the odd-set (blossom) constraints. For sparse constraints, we give a polynomial time separation algorithm if only constraints on all odd sets bounded by a given si...

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Bibliographic Details
Published inDiscrete optimization Vol. 54; p. 100849
Main Authors Hunsaker, Brady, Tovey, Craig
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.11.2024
Subjects
Fractional matching
Odd-set constraints
Perfect matching polytope
Separation
Sparse cuts
Fractional matching
Perfect matching polytope
Sparse cuts
Separation
Odd-set constraints
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Summary:We investigate polytopes intermediate between the fractional matching and the perfect matching polytopes, by imposing a strict subset of the odd-set (blossom) constraints. For sparse constraints, we give a polynomial time separation algorithm if only constraints on all odd sets bounded by a given size (e.g. ≤9+|V|/6) are present. Our algorithm also solves the more general problem of finding a T-cut subject to upper bounds on the cardinality of its defining node set and on its cost. In contrast, regarding dense constraints, we prove that for every 0<α≤12, it is NP-complete to separate over the class of constraints on odd sets of size 2⌊(1+α|V|)/2⌋−1 or ≥α|V|.
ISSN:1572-5286
DOI:10.1016/j.disopt.2024.100849