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 intersectio...

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Published inTheory and Applications of Models of Computation Vol. 12337; pp. 156 - 167
Main Authors Iwamasa, Yuni, Takazawa, Kenjiro
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 2020
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Combinatorial optimization problem with interaction costs
M-convex submodular flow
Matroid congestion game
Valuated independent assignment
Valuated matroid intersection
Online AccessGet full text
ISBN3030592669
9783030592660
ISSN0302-9743
1611-3349
DOI10.1007/978-3-030-59267-7_14

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Summary: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 $$X_1 \cap X_2$$ . 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 $$|X_1 \cap X_2|\le k$$ or $$|X_1 \cap X_2|\ge k$$ ; and designed a new primal-dual algorithm for the case where $$|X_1 \cap X_2|=k$$ . 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 $$\mathrm {M}$$ -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 matroid congestion games and combinatorial optimization problems with interaction costs.
Bibliography:The first author was supported by JSPS KAKENHI Grant Number JP19J01302, Japan. The second author was supported by JSPS KAKENHI Grant Numbers JP16K16012, JP26280004, Japan.
Original Abstract: For two matroids \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_2$$\end{document} with the same ground set V and two cost functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_2$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^V$$\end{document}, we consider the problem of finding bases \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_2$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_2$$\end{document} minimizing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1(X_1)+w_2(X_2)$$\end{document} subject to a certain cardinality constraint on their intersection \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1 \cap X_2$$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X_1 \cap X_2|\le k$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X_1 \cap X_2|\ge k$$\end{document}; and designed a new primal-dual algorithm for the case where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X_1 \cap X_2|=k$$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {M}$$\end{document}-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 matroid congestion games and combinatorial optimization problems with interaction costs.
ISBN:3030592669
9783030592660
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-030-59267-7_14