The Rainbow at the End of the Line - A PPAD Formulation of the Colorful Carathéodory Theorem with Applications

Let C1, ..., Cd+1 be d+ 1 point sets in R, each containing the origin in its convex hull. A subset C of ⋃d+1i=1 Ciis called a colorful choice (or rainbow) for C1, . . . ,Cd+1, if it contains exactly one point from each set Ci. Thecolorful Carathéodory theorem states that there always exists a colorful choice for C1, . . . , Cd+1 that has theorigin in its convex hull. This theorem is very general and can be used to prove several other existence theoremsin high-dimensional discrete geometry, such as the centerpoint theorem or Tverberg’s theorem. The colorfulCarathéodory problem (ColorfulCarathéodory) is the computational problem of finding such a colorfulchoice. Despite several efforts in the past, the computational complexity of ColorfulCarathéodory inarbitrary dimension is still open.We show that ColorfulCarathéodory lies in the intersection of the complexity classes PPAD andPLS. This makes it one of the few geometric problems in PPAD and PLS that are not known to be solvablein polynomial time. Moreover, it implies that the problem of computing centerpoints, computing Tverbergpartitions, and computing points with large simplicial depth is contained in PPAD ∩ PLS. This is the firstnontrivial upper bound on the complexity of these problems.Finally, we show that our PPAD formulation leads to a polynomial-time algorithm for a special case ofColorfulCarathéodory in which we have only two color classes C1 and C2 in d dimensions, each withthe origin in its convex hull, and we would like to find a set with half the points from each color class thatcontains the origin in its convex hull.