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Coverings and delta-coverings

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Integer Programming and Combinatorial Optimization (IPCO 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 920))

Abstract

Given two set systems, (V, F 1) and (V, F 2), the covering problem (resp. delta-covering problem) is to find F 1 ∈ F 1 and F 2 ∈ F 2 maximizing ‖F 1 ∪ F 2‖(resp. ‖F 1 ΔF 2‖). The two problems are equivalent when F 1 and F 2 are the collections of bases of two matroids. We discuss the delta-covering problem when F 1 and F 2 are the collections of feasible sets of two delta-matroids. Applications involve linkings of graphs, Euler tours of 4-regular graphs and nonsingular principal minors of antisymmetric matrices. The delta-covering problem also contains the matroid parity problem, and so it cannot be efficiently solved in general. We state a covering problem related with a new combinatorial structure, called multimatroid, that encompasses the delta-covering problem for delta-matroids. A particular case of the new problem is solved by extending Edmonds' covering theorem for matroids. The solution also implies a theorem of Jackson on pairwise compatible Euler tours of a 4-regular graph.

The paper can be read as an introduction to delta-matroids.

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Authors and Affiliations

  1. Département d'Informatique, Université du Maine, 72017, Le Mans Cedex, France

    André Bouchet

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  1. André Bouchet
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Egon Balas Jens Clausen

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© 1995 Springer-Verlag Berlin Heidelberg

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Bouchet, A. (1995). Coverings and delta-coverings. In: Balas, E., Clausen, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1995. Lecture Notes in Computer Science, vol 920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59408-6_54

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  • DOI: https://doi.org/10.1007/3-540-59408-6_54

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-59408-6

  • Online ISBN: 978-3-540-49245-0

  • eBook Packages: Springer Book Archive

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