Scarf is Ppad-Complete

Scarf’s lemma is one of the fundamental results in combinatorics, originally introduced to study the core of an N -person game. Over the last four decades, the usefulness of Scarf’s lemma has been demonstrated in several important combinatorial problems. However, the complexity of the computational version of Scarf’s lemma (Scarf) is unknown. In this paper, we prove that Scarf is complete for the complexity class PPAD. Hence, there is no polynomial-time algorithm for Scarf unless PPAD ⊆ P.