#### Title

Ranking tournaments with no errors I: Structural description

#### Document Type

Article

#### Publication Date

3-1-2020

#### Abstract

In 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 Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In this first paper we present a structural description of all Möbius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments.

#### Publication Source (Journal or Book title)

Journal of Combinatorial Theory. Series B

#### First Page

264

#### Last Page

294

#### Recommended Citation

Chen, X., Ding, G., Zang, W., & Zhao, Q.
(2020). Ranking tournaments with no errors I: Structural description.* Journal of Combinatorial Theory. Series B**, 141*, 264-294.
https://doi.org/10.1016/j.jctb.2019.08.004