Multivariate analysis of scheduling fair competitions

A 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 Fair Non-Eliminating Tournament (Fair-NET) problem can be used to schedule fair competitions whose infrastructure is known. In the 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 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 ∈ N. We initiate a study of the classical and parameterized complexity of Fair-NET with respect to several central structural parameters motivated by real world scenarios, thereby presenting a comprehensive picture of it.