We consider matroid intersection under the minimum rank oracle model, in which two matroids are accessible only through an oracle that returns the minimum of the ranks of a given set in the two matroids. By Edmonds' theorem, this oracle determines the matroid intersection polytope, although it does not reveal the exchangeability graphs on which classical algorithms rely. Recent work by Bárász, Bérczi, Király, Oki, Yamaguchi, and Yokoi (2025) showed that unweighted matroid intersection is polynomially solvable in this model. In this paper, we present a strongly polynomial-time algorithm for weighted matroid intersection in this model. The key to our algorithm is a polynomial-size extended formulation of tangent cones of common base polytopes that can be constructed using the minimum rank oracle. We also give min-max theorems in terms of the minimum rank function via 1/3-submodularity, and present a strongly polynomial-time algorithm to compute an optimal dual solution.

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