This House Proves that Debating is Harder than Soccer

TL;DR

This paper investigates the computational complexity of determining whether a team can win or qualify in debating tournaments, revealing NP-hardness in some cases and fixed parameter tractability in others, contrasting with simpler sports like football.

Contribution

It proves NP-hardness for debating league outcomes with four teams per game and shows fixed parameter tractability for common tournament scenarios, extending complexity analysis beyond traditional sports.

Findings

NP-hardness for debating leagues with four teams per game

Fixed parameter tractability when remaining rounds are limited

Polynomial time solvability for fixed number of remaining rounds

Abstract

During the last twenty years, a lot of research was conducted on the sport elimination problem: Given a sports league and its remaining matches, we have to decide whether a given team can still possibly win the competition, i.e., place first in the league at the end. Previously, the computational complexity of this problem was investigated only for games with two participating teams per game. In this paper we consider Debating Tournaments and Debating Leagues in the British Parliamentary format, where four teams are participating in each game. We prove that it is NP-hard to decide whether a given team can win a Debating League, even if at most two matches are remaining for each team. This contrasts settings like football where two teams play in each game since there this case is still polynomial time solvable. We prove our result even for a fictitious restricted setting with only three teams per game. On the other hand, for the common setting of Debating Tournaments we show that this problem is fixed parameter tractable if the parameter is the number of remaining rounds $k$. This also holds for the practically very important question of whether a team can still qualify for the knock-out phase of the tournament and the combined parameter $k + b$ where $b$ denotes the threshold rank for qualifying. Finally, we show that the latter problem is polynomial time solvable for any constant $k$ and arbitrary values $b$ that are part of the input.