An Optimal Algorithm for Finding Champions in Tournament Graphs

A tournament graph is a complete directed graph, which can be used to model a round-robin tournament between n players. In this paper, we address the problem of finding a champion of the tournament, also known as Copeland winner, which is a player that wins the highest number of matches. In detail, we aim to investigate algorithms that find the champion by playing a low number of matches. Solving this problem allows us to speed up several Information Retrieval and Recommender System applications, including question answering, conversational search, etc. Indeed, these applications often search for the champion inducing a round-robin tournament among the players by employing a machine learning model to estimate who wins each pairwise comparison. Our contribution, thus, allows finding the champion by performing a low number of model inferences. We prove that any deterministic or randomized algorithm finding a champion with constant success probability requires \Omega (\ell n) comparisons, where \ell is the number of matches lost by the champion. We then present an asymptotically-optimal deterministic algorithm matching this lower bound without knowing \ell, and we extend our analysis to three variants of the problem. Lastly, we conduct a comprehensive experimental assessment of the proposed algorithms on a question answering task on public data. Results show that our proposed algorithms speed up the retrieval of the champion up to 13\times with respect to the state-of-the-art algorithm that perform the full tournament. The identification of the most relevant result from a set of candidates is a crucial task in many Information Retrieval and Recommender System applications including ad-hoc search, conversational search, machine translation, question answering, etc. State-of-the-art solutions solving the task leverage ad-hoc machine learning techniques-developed in a field known as Learning-to-Rank-to estimate the relevance of the set of candidate results and to select the most relevant one. These solutions address the problem in two different ways. From one side, several techniques work by estimating one candida