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

• 成果类型: Article
• 署名作者: Iwamasa, Yuni; Takazawa, Kenjiro
• 署名单位: Kyoto University; Hosei University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-021-01625-2
• 发表日期: 2022
• 页码: 229-256

摘要

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 boolean AND 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 vertical bar X-1 boolean AND X-2 vertical bar = k or vertical bar X-1 boolean AND X-2 vertical bar = k; and designed a new primal-dual algorithm for the case where the constraint is vertical bar X-1 boolean AND X-2 vertical bar = 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 showtheNP-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.