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 $T\backslash F$ contains no cycles (directed). A collection $\mathcal{C}$ of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most $w(e)$ times by members of $\mathcal{C}$. 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.

在这系列的两篇论文中，我们研究了基于体育比赛产生的成对比较而对一组球员进行排名的经典问题，目的是最大程度地减少烦恼的总数，如果较高的烦恼发生排名较高的玩家实际上被排名较低的玩家击败。这个问题可以改写为锦标赛上所谓的最小反馈弧集问题，它出现在各种各样的应用中，并且已经成为广泛研究的主题。在本系列中，我们使用结构驱动和线性规划方法研究NP难题。让$Ť=（V，\mathrm{一种}）$ 积分非负的比赛 $w（Ë）$在每个弧e上。弧的子集F称为反馈弧集，如果$Ť\backslash F$不包含循环（有向）。集合$\mathcal{C}$如果最多使用每个圆弧e，则循环数（允许重复）被称为循环填充$w（Ë）$ 成员的时间 $\mathcal{C}$。如果对于A上定义的每个非负积分函数w，反馈弧集的最小总权重等于循环装箱的最大尺寸，则我们称为T循环Mengerian（CM）。这两篇论文的目的是表明锦标赛是CM，如果它不包含四个Möbius梯子作为子图，则为CM。这样的锦标赛被称为无哥比乌斯锦标赛。在第一篇论文中，我们介绍了所有无Möbius锦标赛的结构性描述，这些锦标赛很大程度上依赖于有关内部2强锦标赛的链定理。