Let C1,…, Cd+i be d + 1 point sets in ℝd, each containing the origin in its convex hull. A subset C of is called a colorful choice (or rainbow) for C1,…, Cd+1, if it contains exactly one point from each set Ci. The colorful Carathéodory theorem states that there always exists a colorful choice for C1,…, Cd+1 that has the origin in its convex hull. This theorem is very general and can be used to prove several other existence theorems in high-dimensional discrete geometry, such as the centerpoint theorem or Tverberg's theorem. The colorful Carathéodory problem (ColorfulCarathéodory) is the computational problem of finding such a colorful choice. Despite several efforts in the past, the computational complexity of ColorfulCarathéodory in arbitrary dimension is still open. We show that ColorfulCarathéodory lies in the intersection of the complexity classes PPAD and PLS. This makes it one of the few geometric problems in PPAD and PLS that are not known to be solvable in polynomial time. Moreover, it implies that the problem of computing centerpoints, computing Tverberg partitions, and computing points with large simplicial depth is contained in PPAD Π PLS. This is the first nontrivial 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 of ColorfulCarathéodory in which we have only two color classes C1 and C2 in d dimensions, each with the origin in its convex hull, and we would like to find a set with half the points from each color class that contains the origin in its convex hull.

Optimization over $l\times m\times n$ integer 3-way tables with given line-sums is NP-hard already for fixed $l=3$, but is polynomial time solvable with both $l,m$ fixed. In the huge version of the problem, the variable dimension $n$ is encoded in binary, with $t$ layer types. It was recently shown that the huge problem can be solved in polynomial time for fixed $t$, and the complexity of the problem for variable $t$ was raised as an open problem. Here we solve this problem and show that the huge table problem can be solved in polynomial time even when the number $t$ of types is variable. The complexity of the problem over 4-way tables with variable $t$ remains open, where all we know is that the associated decision problem is in NP intersect coNP. Our treatment goes through the more general class of huge $n$-fold integer programming problems. We show that huge integer programs over $n$-fold products of totally unimodular matrices can be solved in polynomial time even when the number $t$ of brick types is variable.

The colourful simplicial depth conjecture states that any point in the convex hull of each of $d+1$ sets, or colours, of $d+1$ points in general position in $\mathbb{R}^d$ is contained in at least $d^2+1$ simplices with one vertex from each set. We verify the conjecture in dimension 4 and strengthen the known lower bounds in higher dimensions. These results are obtained using a combinatorial generalization of colourful point configurations called octahedral systems. We present properties of octahedral systems generalizing earlier results on colourful point configurations and exhibit an octahedral system which cannot arise from a colourful point configuration. The number of octahedral systems is also given.