# Optimal matroid bases with intersection constraints: Valuated matroids, M-convex functions, and their applications

For two matroids M_1 and M_2 with the same ground set V and two cost functions w_1 and w_2 on 2^V, we consider the problem of finding bases X_1 of M_1 and X_2 of M_2 minimizing w_1(X_1)+w_2(X_2) subject to a certain cardinality constraint on their intersection X_1 ∩ X_2. Lendl, Peis, and Timmermans (2019) discussed modular cost functions: they reduced the problem to weighted matroid intersection for the case where the cardinality constraint is |X_1 ∩ X_2|< k or |X_1 ∩ X_2|> k; and designed a new primal-dual algorithm for the case where the constraint is |X_1 ∩ X_2|=k. The aim of this paper is to generalize the problems to have nonlinear convex cost functions, and to comprehend them from the viewpoint of discrete convex analysis. We prove that each generalized problem can be solved via valuated independent assignment, valuated matroid intersection, or M-convex submodular flow, to offer a comprehensive understanding of weighted matroid intersection with intersection constraints. We also show the NP-hardness of some variants of these problems, which clarifies the coverage of discrete convex analysis for those problems. Finally, we present applications of our generalized problems in the recoverable robust matroid basis problem, matroid congestion games, and combinatorial optimization problems with interaction costs.

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