On The Structure of Parametric Tournaments with Application to Ranking from Pairwise Comparisons
Authors: Vishnu Veerathu, Arun Rajkumar

 We consider the classical problem of finding the minimum feedback arc set on tournaments (MFAST). The problem is NP-hard in general and we study it for important classes of tournaments that arise naturally in the problem of learning to rank from pairwise comparisons. Specifically, we consider tournaments classes that arise out of parametric preference matrices that can lead to cyclic preference relations. We investigate their structural properties via forbidden sub tournament configurations.  Towards this, we introduce \emph{Tournament Dimension} - a combinatorial parameter that characterizes the size of a forbidden configuration for rank $r$ tournament classes i.e., classes that arise out pairwise preference matrices which lead to rank $r$ skew-symmetric matrices under a suitable link function. Our main result is a polynomial-time algorithm - \texttt{Rank2Rank} - that solves the MFAST problem for the rank $2$ tournament class. This is achieved via a  geometric characterization that relies on our explicit construction of a forbidden configuration for this class.   Building on our understanding of the rank-$2$ tournament class, we propose a very general and flexible parametric pairwise preference model called the local-global model which subsumes the popular Bradley-Terry-Luce/Thurstone classes to capture locally cyclic as well as globally acyclic preference relations. We develop a polynomial-time algorithm - \texttt{BlockRank2Rank}- to solve the MFAST problem on the associated Block-Rank $2$ tournament class.  As an application, we study the problem of learning to rank from pairwise comparisons under the proposed local-global preference model. Exploiting our structural characterization, we propose  \texttt{PairwiseBlockRank} - a pairwise ranking algorithm for this class. We show sample complexity bounds of \texttt{PairwiseBlockRank}  to learn a good ranking under the proposed model.  Finally, we conduct experiments on synthetic and real-world datasets to show the efficacy of the proposed algorithm.