Weighted Exchange Distance of Basis Pairs

Two pairs of disjoint bases P 1 = ( R 1, B 1 ) and P 2 = ( R 2, B 2 ) of a matroid M are called equivalent if P 1 can be transformed into P 2 by a series of symmetric exchanges. In 1980, White conjectured that such a sequence always exists whenever R 1 ∪ B 1 = R 2 ∪ B 2. A strengthening of the conjecture was proposed by Hamidoune, stating that the minimum length of an exchange is at most the rank of the matroid. We propose a weighted variant of Hamidoune’s conjecture, where the weight of an exchange depends on the weights of the exchanged elements. We prove the conjecture for several matroid classes: strongly base orderable matroids, split matroids, graphic matroids of wheels, and spikes.