On matroid intersections

Abstract

This paper exploits and extends results of Edmonds, Cunningham, Cruse and McDiarmid on matroid intersections. Letr 1 andr 2 be rank functions of two matroids defined on the same setE. For everySE, letr 12(S) be the largest cardinality of a subset ofS independent in both matroids, 0≦kr 12(E)−1. It is shown that, ifc is nonnegative and integral, there is ay: 2EZ + which maximizes\(\sum\limits_S {(k - r_{12} (E - S))y(S)} \) and\(\sum\limits_S {(k + 1 - r_{12} (E - S))y(S)} \), subject toy≧0, ∀jE,\(\sum\limits_{S \mathrel\backepsilon j} {y(S) \leqq c_j } \).

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Gröflin, H., Hoffman, A.J. On matroid intersections. Combinatorica 1, 43–47 (1981). https://doi.org/10.1007/BF02579175

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AMS subject classification (1980)

  • 05 B 35