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Ranking tournaments with no errors I: Structural description
Chen, Xujin1,2; Ding, Guoli3; Zang, Wenan4; Zhao, Qiulan5
AbstractIn this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem on tournaments, which arises in a rich variety of applications and has been a subject of extensive research. In this series we study this NP-hard problem using structure-driven and linear programming approaches. Let T = (V, A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set if T\F contains no cycles (directed). A collection C of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e) times by members of C. We call T cycle Mengerian (CM) if, for every nonnegative integral function w defined on A, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. The purpose of these two papers is to show that a tournament is CM iff it contains none of four Mobius ladders as a subgraph; such a tournament is referred to as Mobius-free. In this first paper we present a structural description of all Mobius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments. (C) 2019 Elsevier Inc. All rights reserved.
KeywordTournament Feedback arc set Cycle packing Minimax relation Characterization
Indexed BySCI
Funding ProjectNSF of China[11801266] ; NSF of China[11531014] ; NSF[DMS-1500699] ; Research Grants Council of Hong Kong
WOS Research AreaMathematics
WOS SubjectMathematics
WOS IDWOS:000508288900012
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Document Type期刊论文
Corresponding AuthorZhao, Qiulan
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China
2.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
3.Louisiana State Univ, Math Dept, Baton Rouge, LA 70803 USA
4.Univ Hong Kong, Dept Math, Hong Kong, Peoples R China
5.Nanjing Univ, Dept Math, Nanjing 210093, Peoples R China
Recommended Citation
GB/T 7714
Chen, Xujin,Ding, Guoli,Zang, Wenan,et al. Ranking tournaments with no errors I: Structural description[J]. JOURNAL OF COMBINATORIAL THEORY SERIES B,2020,141:264-294.
APA Chen, Xujin,Ding, Guoli,Zang, Wenan,&Zhao, Qiulan.(2020).Ranking tournaments with no errors I: Structural description.JOURNAL OF COMBINATORIAL THEORY SERIES B,141,264-294.
MLA Chen, Xujin,et al."Ranking tournaments with no errors I: Structural description".JOURNAL OF COMBINATORIAL THEORY SERIES B 141(2020):264-294.
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