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 fixe...
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Zusammenfassung: | 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 which must necessarily be union of flip
classes, a novel way to partition the set of all tournaments. We further
characterise 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. 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 \cite{forster}, we show a lower bound of $\mathcal{O}(\sqrt{n})$ 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 upper bound of sign-rank of matrices. |
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DOI: | 10.48550/arxiv.2110.05188 |