Computing the minimal covering set

We present the first polynomial-time algorithm for Computing the minimal covering set of a (weak) tournament. The algorithm draws upon a linear programming formulation of a subset of the minimal covering set known as the essential set. On the other hand, we show that no efficient algorithm exists for two variants of the minimal covering set - the minimal upward covering set and the minimal downward covering set - unless P equals NP. Finally, we observe a strong relationship between von Neumann-Morgenstern stable sets and upward covering on the one hand, and the Banks set and downward covering on the other. (C) 2008 Elsevier B.V. All rights reserved.