Ranking tournaments with no errors I: Structural description

Xujin Chen, Guoli Ding, Wenan Zang, Qiulan Zhao

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)<math><mi is="true">T</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mo stretchy="false" is="true">(</mo><mi is="true">V</mi><mo is="true">,</mo><mi is="true">A</mi><mo stretchy="false" is="true">)</mo></math> be a tournament with a nonnegative integral weight w(e)<math><mi is="true">w</mi><mo stretchy="false" is="true">(</mo><mi is="true">e</mi><mo stretchy="false" is="true">)</mo></math> on each arc e. A subset F of arcs is called a feedback arc set if T\F<math><mi is="true">T</mi><mo is="true">\</mo><mi is="true">F</mi></math> contains no cycles (directed). A collection C<math><mi mathvariant="script" is="true">C</mi></math> of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e)<math><mi is="true">w</mi><mo stretchy="false" is="true">(</mo><mi is="true">e</mi><mo stretchy="false" is="true">)</mo></math> times by members of C<math><mi mathvariant="script" is="true">C</mi></math>. 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.