Let C 1 , ..., C d+1 be d + 1 point sets in R d , each containing the origin in its convex hull. A subset C of d+1 i=1 C i is called a colorful choice (or rainbow) for C 1 , . . . , C d+1 , if it contains exactly one point from each set C i . The colorful Carathéodory theorem states that there always exists a colorful choice for C 1 , . . . , C d+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 ∩ 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 C 1 and C 2 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.

Let S be a planar n-point set. A triangulation for S is a maximal plane straight-line graph with vertex set S. The Voronoi diagram for S is the subdivision of the plane into cells such that all points in a cell have the same nearest neighbor in S. Classically, both structures can be computed in O(n log n) time and O(n) space. We study the situation when the available workspace is limited: given a parameter s ∈ {1, . . . , n}, an s-workspace algorithm has readonly access to an input array with the points from S in arbitrary order, and it may use only O(s) additional words of Θ(log n) bits for reading and writing intermediate data. The output should then be written to a write-only structure. We describe a deterministic s-workspace algorithm for computing an arbitrary triangulation of S in time O(n 2 /s + n log n log s) and a randomized s-workspace algorithm for finding the Voronoi diagram of S in expected time O((n 2 /s) log s + n log s log * s). * WS and PS were supported in part by DFG Grants MU 3501/1 and MU 3501/2. YS was supported by the DFG within the research training group "Methods for Discrete Structures" (GRK 1408).The exact setting may vary, but there is a common theme: the input resides in read-only memory, the output must be written to a write-only structure, and we can use O(s) additional variables to find the solution (for a parameter s). The goal is to design algorithms whose running time decreases as s grows, giving a time-space trade-off [23]. One of the first problems considered in this model is sorting [19,20]. Here, the time-space product is known to be Ω(n 2 ) [8], and matching upper bounds for the case b ∈ Ω(log n) ∩ O(n/ log n) were obtained by Pagter and Rauhe [21] (b denotes the available workspace in bits).Our current notion of memory constrained algorithms was introduced to computational geometry by Asano et al. [4], who showed how to compute many classic geometric structures with O(1) workspace (related models were studied before [9]). Later, time-space trade-offs were given for problems on simple polygons, e.g., shortest paths [1], visibility [6], or the convex hull of the vertices [5].We consider a model in which the set S of n points is in an array such that random access to each input point is possible, but we may not change or even reorder the input. Additionally, we have O(s) variables (for a parameter s ∈ {1, . . . , n}). We assume that each variable or pointer contains a data word of Θ(log n) bits. Other than this, the model allows the usual word RAM operations. In this setting we study two problems: computing an arbitrary triangulation for S and computing the Voronoi diagram VD(S) for S. Since the output cannot be stored explicitly, the goal is to report the edges of the triangulation or the vertices of VD(S) successively, in no particular order. Dually, the latter goal may be phrased in terms of Delaunay triangulations. We focus on Voronoi diagrams, as they lead to a more natural presentation.Both problems can be solved in O(n 2 ) time with O(1) workspace [4] or in O(n log n) ...

Let P be a d-dimensional n-point set. A partition T of P is called a Tverberg partition if the convex hulls of all sets in T intersect in at least one point. We say that T is t-tolerant if it remains a Tverberg partition after deleting any t points from P . Soberón and Strausz proved that there is always a t-tolerant Tverberg partition with ⌈n/(d+1)(t+1)⌉ sets. However, no nontrivial algorithms for computing or approximating such partitions have been presented so far. For d ≤ 2, we show that the Soberón-Strausz bound can be improved, and we show how the corresponding partitions can be found in polynomial time. For d ≥ 3, we give the first polynomial-time approximation algorithm by presenting a reduction to the regular Tverberg problem (with no tolerance). Finally, we show that it is coNP-complete to determine whether a given Tverberg partition is t-tolerant.

Let C1, . . . , C d+1 ⊂ R d be d + 1 point sets, each containing the origin in its convex hull. We call these sets color classes, and we call a sequence p1, . . . , p d+1 with pi ∈ Ci, for i = 1, . . . , d + 1, a colorful choice. The colorful Carathéodory theorem guarantees the existence of a colorful choice that also contains the origin in its convex hull. The computational complexity of finding such a colorful choice (ColorfulCarathéodory) is unknown. This is particularly interesting in the light of polynomial-time reductions from several related problems, such as computing centerpoints, to ColorfulCarathéodory.We define a novel notion of approximation that is compatible with the polynomial-time reductions to Col-orfulCarathéodory: a sequence that contains at most k points from each color class is called a k-colorful choice. We present an algorithm that for any fixed ε > 0, outputs an εd -colorful choice containing the origin in its convex hull in polynomial time.Furthermore, we consider a related problem of ColorfulCarathéodory: in the nearest colorful polytope problem (Ncp), we are given sets C1, . . . , Cn ⊂ R d that do not necessarily contain the origin in their convex hulls. The goal is to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing a local optimum for Ncp is PLS-complete, while computing a global optimum is NP-hard.

We consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes. We may preprocess P to obtain a label and a routing table for each vertex of P . Then, we must be able to route a data packet between any two vertices p and q of P , where each step must use only the label of the target node q and the routing table of the current node.For any fixed ε > 0, we present a routing scheme that always achieves a routing path whose length exceeds the shortest path by a factor of at most 1 + ε. The labels have O(log n) bits, and the routing tables are of size O((ε −1 + h) log n). The preprocessing time is O(n 2 log n). It can be improved to O(n 2 ) for simple polygons.