A Theory of Tournament Representations

Real-world tournaments are almost always intransitive. Recent works have noted that parametric models which assume $d$ dimensional node representations can effectively model intransitive tournaments. However, nothing is known about the structure of the class of tournaments that arise out of any fixed $d$ dimensional representations. In this work, we develop a novel theory for understanding parametric tournament representations. Our first contribution is to structurally characterize the class of tournaments that arise out of $d$ dimensional representations. We do this by showing that these tournament classes have forbidden configurations that must necessarily be a union of flip classes, a novel way to partition the set of all tournaments. We further characterize rank $2$ tournaments completely by showing that the associated forbidden flip class contains just $2$ tournaments. Specifically, we show that the rank $2$ tournaments are equivalent to locally transitive tournaments. This insight allows us to show that the minimum feedback arc set problem on this tournament class can be solved using the standard Quicksort procedure. We also exhibit specific forbidden configurations for rank $4$ tournaments. For a general rank $d$ tournament class, we show that the flip class associated with a coned-doubly regular tournament of size $\mathcal{O}(\sqrt{d})$ must be a forbidden configuration. To answer a dual question, using a celebrated result of Froster, we show a lower bound of $\Theta(\sqrt{n})$ on the minimum dimension needed to represent all tournaments on $n$ nodes. For any given tournament, we show a novel upper bound on the smallest representation dimension that depends on the least size of the number of unique nodes in any feedback arc set of the flip class associated with a tournament. We show how our results also shed light on the upper bound of sign-rank of matrices.