Multivariate Analysis of Scheduling Fair Competitions

A \emph{fair competition}, based on the concept of envy-freeness, is a non-eliminating competition where each contestant (team or individual player) may not play against all other contestants, but the total difficulty for each contestant is the same: the sum of the initial rankings of the opponents for each contestant is the same. Similar to other non-eliminating competitions like the Round-robin competition or the Swiss-system competition, the winner of the fair competition is the contestant who wins the most games. The {\sc Fair Non-Eliminating Tournament} ({\sc Fair-NET}) problem can be used to schedule fair competitions whose infrastructure is known. In the {\sc Fair-NET} problem, we are given an infrastructure of a tournament represented by a graph $G$ and the initial rankings of the contestants represented by a multiset of integers $S$. The objective is to decide whether $G$ is \emph{$S$-fair}, i.e., there exists an assignment of the contestants to the vertices of $G$ such that the sum of the rankings of the neighbors of each contestant in $G$ is the same constant $k\in\mathbb{N}$. We initiate a study of the classical and parameterized complexity of {\sc Fair-NET} with respect to several central structural parameters motivated by real world scenarios, thereby presenting a comprehensive picture of it.