Complexity of Scarf’s Lemma and Related Problems

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 seeking stable solutions ([12], [1] [2], [15]). However, the complexity of the computational version of Scarf’s lemma (Scarf) remained open. In this paper, we prove that Scarf is complete for the complexity class PPAD. This proves that Scarf is as hard as the computational versions of Brouwer’s fixed point theorem and Sperner’s lemma. Hence, there is no polynomial-time algorithm for Scarf unless PPAD ⊆ P. We also show that fractional stable paths problem and finding strong fractional kernels in digraphs are PPAD-hard.