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 everyS ⊂E, letr ₁₂(S) be the largest cardinality of a subset ofS independent in both matroids, 0≦k≦r ₁₂(E)−1. It is shown that, ifc is nonnegative and integral, there is ay: 2^E →Z ⁺ 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, ∀j∈E,$\sum\limits_{S \mathrel\backepsilon j} {y(S) \leqq c_j }$.