Abstract Tverberg’s theorem states that for any $$k\ge 2$$ k ≥ 2 and any set $$P\subset {\mathbb {R}}^d$$ P ⊂ R d of at least $$(d+1)(k-1)+1$$ ( d + 1 ) ( k - 1 ) + 1 points in d dimensions, we can partition P into k subsets whose convex hulls have a non-empty intersection. The associated search problem of finding the partition lies in the complexity class $$\text {CLS} = \text {PPAD} \cap \text {PLS}$$ CLS = PPAD ∩ PLS , but no hardness results are known. In the colorful Tverberg theorem, the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Bárány, and Mustafa (SODA 2019) gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of P. The argument is constructive, but does not result in a polynomial-time algorithm. We present a deterministic algorithm that finds for any n-point set $$P\subset {\mathbb {R}}^d$$ P ⊂ R d and any $$k\in \{2,\dots ,n\}$$ k ∈ { 2 , ⋯ , n } in $$O(nd\lceil {\log k}\rceil )$$ O ( n d ⌈ log k ⌉ ) time a k-partition of P such that there is a ball of radius $$O((k/\sqrt{n}){\text {diam}}(P))$$ O ( ( k / n ) diam ( P ) ) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, our result provides a remarkably efficient and simple new notion of approximation. Our main contribution is to generalize Sarkaria’s method (Israel Journal Math., 1992) to reduce the Tverberg problem to the colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem.

Tverberg’s theorem states that for any k ≥ 2 and any set P ⊂ ℝ^d of at least (d + 1)(k - 1) + 1 points, we can partition P into k subsets whose convex hulls have a non-empty intersection. The associated search problem lies in the complexity class PPAD ∩ PLS, but no hardness results are known. In the colorful Tverberg theorem, the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Bárány, and Mustafa [SODA 2019] gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of P. The argument is constructive, but does not result in a polynomial-time algorithm. We present a deterministic algorithm that finds for any n-point set P ⊂ ℝ^d and any k ∈ {2, … , n} in O(nd ⌈log k⌉) time a k-partition of P such that there is a ball of radius O((k/√n)diam(P)) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, and that there are no approximation algorithms that are truly polynomial in any dimension, our result provides a remarkably efficient and simple new notion of approximation. Our main contribution is to generalize Sarkaria’s method [Israel Journal Math., 1992] to reduce the Tverberg problem to the Colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem.

The classic Ham-Sandwich theorem states that for any $d$ measurable sets in $\mathbb{R}^d$, there is a hyperplane that bisects them simultaneously. An extension by Bárány, Hubard, and Jerónimo [DCG 2008] states that if the sets are convex and \emph{well-separated}, then for any given $α_1, \dots, α_d \in [0, 1]$, there is a unique oriented hyperplane that cuts off a respective fraction $α_1, \dots, α_d$ from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the \emph{$α$-Ham-Sandwich theorem}. They gave an algorithm to find the hyperplane in time $O(n (\log n)^{d-3})$, where $n$ is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]). Recently, Fearley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called \emph{Unique End-of-Potential Line} (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the $α$-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the $P$-matrix linear complementarity problem.

Čech complexes are useful simplicial complexes for computing and analyzing topological features of data that lies in Euclidean space. Unfortunately, computing these complexes becomes prohibitively expensive for large-sized data sets even for medium-to-low dimensional data. We present an approximation scheme for (1 + ε)-approximating the topological information of the Čech complexes for n points in ℝd, for ε ∊ (0, 1]. Our approximation has a total size of for constant dimension d, improving all the currently available (1 + ε)-approximation schemes of simplicial filtrations in Euclidean space. Perhaps counter-intuitively, we arrive at our result by adding additional sample points to the input. We achieve a bound that is independent of the spread of the point set by pre-identifying the scales at which the Čech complexes changes and sampling accordingly.

Classical methods to model topological properties of point clouds, such as the Vietoris–Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in $$\mathbb {R}^d$$ , we obtain a O(d)-approximation whose k-skeleton has size $$n2^{O(d \log k)}$$ per scale and $$n2^{O(d\log d)}$$ in total over all scales. In conjunction with dimension reduction techniques, our approach yields a $$O(\mathrm {polylog} (n))$$ -approximation of size $$n^{O(1)}$$ for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry. Building on the same geometric concept, we also present a lower bound result on the size of an approximation: we construct a point set for which every $$(1+\varepsilon )$$ -approximation of the Čech filtration has to contain $$n^{\Omega (\log \log n)}$$ features, provided that $$\varepsilon <\frac{1}{\log ^{1+c} n}$$ for $$c\in (0,1)$$ .

Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes constitutes an expensive task because of a combinatorial explosion in the complex size. For n points in R^d, we present a scheme to construct a 4.24-approximation of the multi-scale filtration of the Rips complex in the L-infinity metric, which extends to a O(d^{0.25})-approximation of the Rips filtration for the Euclidean case. The k-skeleton of the resulting approximation has a total size of n2^{O(d log k)}. The scheme is based on the integer lattice and on the barycentric subdivision of the d-cube.