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

• Published in: Mathematical programming Vol. 194; no. 1-2; pp. 229 - 256
• Main Authors: Iwamasa, Yuni, Takazawa, Kenjiro
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2022; Springer; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorial analysis; Combinatorics; Convex analysis; Cost function; Full Length Paper; Intersections; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Theoretical; 90C27 (Mathematical programming: Combinatorial optimization; Combinatorial optimization problem with interaction costs; Congestion game; Valuated independent assignment; 68Q25 (Theory of computing: Analysis of algorithms and problem complexity); M-convex submodular flow; Recoverable robust matroid basis problem; Valuated matroid intersection
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-021-01625-2

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 . For this problem, Lendl et al. (Matroid bases with cardinality constraints on the intersection, arXiv:1907.04741v2 , 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, combinatorial optimization problems with interaction costs, and matroid congestion games.