Matroid bases with cardinality constraints on the intersection

Author:

Lendl StefanORCID,Peis Britta,Timmermans Veerle

Abstract

AbstractGiven two matroids $$\mathcal {M}_{1} = (E, \mathcal {B}_{1})$$ M 1 = ( E , B 1 ) and $$\mathcal {M}_{2} = (E, \mathcal {B}_{2})$$ M 2 = ( E , B 2 ) on a common ground set E with base sets $$\mathcal {B}_1$$ B 1 and $$\mathcal {B}_2$$ B 2 , some integer $$k \in \mathbb {N}$$ k N , and two cost functions $$c_{1}, c_{2} :E \rightarrow \mathbb {R}$$ c 1 , c 2 : E R , we consider the optimization problem to find a basis $$X \in \mathcal {B}_{1}$$ X B 1 and a basis $$Y \in \mathcal {B}_{2}$$ Y B 2 minimizing the cost $$\sum _{e\in X} c_1(e)+\sum _{e\in Y} c_2(e)$$ e X c 1 ( e ) + e Y c 2 ( e ) subject to either a lower bound constraint $$|X \cap Y| \le k$$ | X Y | k , an upper bound constraint $$|X \cap Y| \ge k$$ | X Y | k , or an equality constraint $$|X \cap Y| = k$$ | X Y | = k on the size of the intersection of the two bases X and Y. The problem with lower bound constraint turns out to be a generalization of the Recoverable Robust Matroid problem under interval uncertainty representation for which the question for a strongly polynomial-time algorithm was left as an open question in Hradovich et al. (J Comb Optim 34(2):554–573, 2017). We show that the two problems with lower and upper bound constraints on the size of the intersection can be reduced to weighted matroid intersection, and thus be solved with a strongly polynomial-time primal-dual algorithm. We also present a strongly polynomial, primal-dual algorithm that computes a minimum cost solution for every feasible size of the intersection k in one run with asymptotic running time equal to one run of Frank’s matroid intersection algorithm. Additionally, we discuss generalizations of the problems from matroids to polymatroids, and from two to three or more matroids. We obtain a strongly polynomial time algorithm for the recoverable robust polymatroid base problem with interval uncertainties.

Funder

Austrian Science

Publisher

Springer Science and Business Media LLC

Subject

General Mathematics,Software

Reference18 articles.

1. Büsing, C.: Recoverable Robustness in Combinatorial Optimization. Cuvillier Verlag, New York (2011)

2. Chassein, A., Goerigk, M.: On the complexity of min-max-min robustness with two alternatives and budgeted uncertainty. Discrete Appl. Math. 3, 76109 (2020)

3. Cunningham, W.H., Geelen, J.F.: The optimal path-matching problem. Combinatorica 17(3), 315–337 (1997)

4. Edmonds, J.: Submodular functions, matroids and certain polyhedra, combinatorial structures and their applications (R. Guy, H. Hanani, N. Sauer and J. Schönheim, eds.). Gordon and Breach pp. 69–87 (1970)

5. Frank, A.: A weighted matroid intersection algorithm. J. Algorithms 2(4), 328–336 (1981)

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Robust Spanning Tree Problems;International Series in Operations Research & Management Science;2024

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3