Football Elimination Is Hard to Decide Under the 3-Point-Rule

Abstract

The “baseball elimination problem” is a classic problem which has already been considered from the computational point of view in the 1960’s. At some stage of the baseball season, there is a set of games which have already been played and there is another set of remaining games. The problem consists in determining for a given team whether or not they are already “eliminated”, i.e., whether they can no longer become champions. Early solutions proposed a network flow approach which resulted in polynomial time algorithms. The interest in this kind of elimination problem was recently revived by Wayne [4] who proved an interesting threshold property which allows one to compute all eliminated teams simultaneously. Namely, there is a constant W* such that a team is eliminated if and only if it can no longer obtain W* or more points. Wayne also describes an algorithm for computing the threshold W* in polynomial time. Gusfield and Martel [2] have generalized the proof of the existence of a threshold to a more general setting which includes European football, where the “3-point-rule” is in effect, i.e., 3 points are awarded for a win and 1 point is awarded for a tie.

In this paper, we show that determining the elimination of a European football team under the 3-point-rule is $\mathcal{N}\mathcal{P}$-complete. As a consequence, the generalized threshold result of Gusfield and Martel is of no use for the European football system since computing the corresponding threshold value is hard if $\mathcal{P} \ne \mathcal{N}\mathcal{P}$. We also show that the elimination problem is still $\mathcal{N}\mathcal{P}$-complete if all teams have at most three remaining games each while the problem can be solved in polynomial time if each team has at most two remaining games.