Inverse Problems of Matroid Intersection

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In this paper we study the inverse problem of matroid intersection: Two matroids M 1 = (E, $${\mathcal{I}}$$ 1) and M 2 = (E, $${\mathcal{I}}$$ 2), their intersection B, and a weight function w on E are given. We try to modify weight w, optimally and with bounds, such that B becomes a maximum weight intersection under the modified weight. It is shown in this paper that the problem can be formulated as a combinatorial linear program and can be further transformed into a minimum cost circulation problem. Hence it can be solved by strongly polynomial time algorithms. We also give a necessary and sufficient condition for the feasibility of the problem. Finally we extend the discussion to the version of the problem with Multiple Intersections.