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 . For this problem,...

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Veröffentlicht in:	Mathematical programming Jg. 194; H. 1-2; S. 229 - 256
Hauptverfasser:	Iwamasa, Yuni, Takazawa, Kenjiro
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2022 Springer Springer Nature B.V
Schlagworte:	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
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Abstract	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.
AbstractList	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. For two matroids M1 and M2 with the same ground set V and two cost functions w1 and w2 on 2V, we consider the problem of finding bases X1 of M1 and X2 of M2 minimizing w1(X1)+w2(X2) subject to a certain cardinality constraint on their intersection X1∩X2. 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 \|X1∩X2\|≤k or \|X1∩X2\|≥k; and designed a new primal-dual algorithm for the case where the constraint is \|X1∩X2\|=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. For two matroids [Formula omitted] and [Formula omitted] with the same ground set V and two cost functions [Formula omitted] and [Formula omitted] on [Formula omitted], we consider the problem of finding bases [Formula omitted] of [Formula omitted] and [Formula omitted] of [Formula omitted] minimizing [Formula omitted] subject to a certain cardinality constraint on their intersection [Formula omitted]. 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 [Formula omitted] or [Formula omitted]; and designed a new primal-dual algorithm for the case where the constraint is [Formula omitted]. 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 [Formula omitted]-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.
Audience	Academic
Author	Iwamasa, Yuni Takazawa, Kenjiro
Author_xml	– sequence: 1 givenname: Yuni orcidid: 0000-0002-6794-3543 surname: Iwamasa fullname: Iwamasa, Yuni email: iwamasa@i.kyoto-u.ac.jp organization: Department of Communications and Computer Engineering, Graduate School of Informatics, Kyoto University – sequence: 2 givenname: Kenjiro orcidid: 0000-0002-7662-7374 surname: Takazawa fullname: Takazawa, Kenjiro organization: Department of Industrial and Systems Engineering, Faculty of Science and Engineering, Hosei University
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Cites_doi	10.1137/S1052623499352012 10.1007/s11590-016-1057-x 10.1016/S0167-5060(08)70734-9 10.1287/moor.7.3.334 10.1006/aima.1996.0084 10.1016/0196-6774(81)90032-8 10.1007/s004930050047 10.1145/351827.384253 10.1002/net.3230120102 10.1007/978-3-540-76796-1_11 10.1287/moor.24.1.95 10.1016/0893-9659(90)90009-Z 10.1145/1455248.1455249 10.1137/16M1107450 10.1007/978-3-319-00795-3_22 10.1016/0001-8708(92)90028-J 10.1007/s10107-016-1053-z 10.1007/s10878-016-0089-6 10.1145/28869.28874 10.1007/s10107-004-0562-3 10.1137/S0895480195280009 10.1093/acprof:oso/9780198566946.001.0001 10.1007/BF01737559 10.1016/j.disopt.2019.03.004 10.1007/BF01681329 10.1006/game.1996.0044 10.1137/1.9780898718508 10.1007/s10878-019-00435-9 10.1137/S0895480195279994
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Keywords	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
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Snippet	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... For two matroids [Formula omitted] and [Formula omitted] with the same ground set V and two cost functions [Formula omitted] and [Formula omitted] on [Formula... For two matroids M1 and M2 with the same ground set V and two cost functions w1 and w2 on 2V, we consider the problem of finding bases X1 of M1 and X2 of M2...
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SubjectTerms	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
Title	Optimal matroid bases with intersection constraints: valuated matroids, M-convex functions, and their applications
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