2D-TUCKER is PPAD-complete

Abstract

Tucker's lemma states that if we triangulate the unit disc centered at the origin and color the vertices with {1,-1,2,-2} in an antipodal way (if |z|=1, then the sum of the colors of z and -z is zero), then there must be an edge for which the sum of the colors of its endpoints is zero. But how hard is it to find such an edge? We show that if the triangulation is exponentially large and the coloring is determined by a deterministic Turing-machine, then this problem is PPAD-complete which implies that there is not too much hope for a polynomial algorithm. © 2009 Springer-Verlag Berlin Heidelberg.