No-dimensional Tverberg Theorems and Algorithms

07/09/2019 ∙ by Aruni Choudhary, et al. ∙ 0

Tverberg's theorem is a classic result in discrete geometry. It states that for any integer k > 2 and any finite d-dimensional point 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 computational problem of finding such a partition lies in the complexity class PPAD∩PLS, but no hardness results are known. Tverberg's theorem also has a colorful variant: the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, i.e., sets in which each color appears exactly once such that the convex hulls of the sets intersect. Recently, Adiprasito, Barany, and Mustafa [SODA 2019] proved a no-dimensional version of Tverberg's theorem, in which the convex hulls of the sets in the partition may intersect in an approximate fashion, relaxing the requirement on the cardinality of P. The argument is constructive, but it does not result in a polynomial-time algorithm. We present an alternative proof for a no-dimensional Tverberg theorem that leads to an efficient algorithm to find the partition. More specifically, we show an deterministic algorithm that finds for any set P ⊆R^d of n points and any k ∈{2, ..., n} in O(nd log k ) time a partition of P into k subsets such that there is a ball of radius O(k/√(n)diam(P)) intersecting the convex hull of each subset. A similar result holds also for the colorful version. To obtain our result, we generalize Sarkaria's tensor product constructions [Israel Journal Math., 1992] that reduces the Tverberg problem to the Colorful Caratheodory problem. By carefully choosing the vectors used in the tensor products, we implement the reduction in an efficient manner.

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1 Introduction

In 1921, Radon [13] proved a highly influential theorem in convex geometry: given a set of at least points in , it is always possible to split into two non-empty sets whose convex hulls intersect. In 1966, Helge Tverberg [17] generalized Radon’s theorem to allow for more sets in the partition. Specifically, he showed that any finite point set of cardinality at least can be split into sets whose convex hulls have a non-empty intersection, i.e., , where denotes the convex hull.

By now, several alternative proofs of Tverberg’s original result are known, e.g., [18, 19, 14, 10, 15, 6, 3]. Perhaps the most elegant proof is due to Sarkaria [15], with later simplifications by Bárány and Onn [6] and by Aroch et al. [3]. This proof proceeds by a reduction to the Colorful Carathéodory Theorem, another celebrated result in discrete geometry. The theorem states that given finite -dimensional point sets that have a common point in their convex hulls , there is a traversal , such that contains . Sarkaria’s proof [15] proceeds by lifting the original points of the Tverberg instance into higher dimensions using a tensor product, and then uses the existence of the colorful Carathéodory traversal to obtain a Tverberg partition for the original point set.

On the computational side of things, a Radon partition is easy to compute by solving linear equations. On the other hand, finding Tverberg partitions is not straightforward. Since a Tverberg partition is guaranteed to exist if the cardinality of is large enough, finding such a partition is a total search problem. In fact, the problem of computing a colorful Carathéodory traversal lies in the complexity class  [11, 9], but no better upper bound on the difficulty of the problem is known. Since Sarkaria’s proof can be interpreted as a polynomial-time reduction from the problem of finding a Tverberg partition to the problem of finding a colorful traversal, the same upper bound applies to finding Tverberg partitions. Again, as of now we do not know better upper bounds for the general problem. Miller and Sheehy [10] and Mulzer and Werner [12] provided algorithms for finding approximate Tverberg partitions, computing a partition into fewer sets than is guaranteed by Tverberg’s theorem in time that is linear in , but quasi-polynomial in the dimension.

Tverberg’s theorem also admits a colorful variant that was first conjectured by Bárány and Larman [5]. The conjecture states that given point sets , each interpreted as a different color, and each set having size at most , there exist pairwise-disjoint colorful sets (i.e., each set contains at most one point from each ) such that . Bárány and Larman [5] proved the conjecture for and arbitrary , and for and arbitrary . The first proof for the general case was given by Živaljević and Vrećica [20] through topological arguments. Using another topological argument, Blagojevič, Matschke, and Ziegler [7] showed that (i) if is prime, then ; and (ii) if is not prime, then . These are the best known bounds for arbitrary . Later Matoušek, Tancer, and Wagner [8] gave a constructive geometric proof that is inspired by the proof of Blagojevič, Matschke, and Ziegler [7].

More recently, Soberón [16] showed that if more color classes are available, then the conjecture holds for any . More precisely, for with , each of size , there exist colorful sets whose convex hulls intersect. Moreover, there is at least one point in the common intersection such that the coefficients of its convex combination are the same for each colorful set in the partition. The proof makes use of Sarkaria’s tensor product construction.

Recent developments.

Recently Adiprasito, Bárány, and Mustafa [1] established a relaxed version of the Colorful Carathéodory Theorem [4]. This version allows for (relaxed) traversals of arbitrary size , with a guarantee that the traversal is close to the common point . Adiprasito, Bárány, and Mustafa [1] also proved a relaxed variant of Colorful Tverberg theorem [5]. This also gives a relaxation for Tverberg’s theorem [17] that allows arbitrary-sized partitions. The authors refer to these results as no-dimensional versions of the respective classic theorems, since the dependence on the ambient dimension is relaxed. Both results were proven using averaging arguments. The argument for the no-dimensional Colorful Carathéodory also gives an efficient algorithm to find a traversal that is close to . However, the arguments for the no-dimensional Tverberg results do not give a polynomial-time algorithm for finding the Tverberg partitions.

Contributions.

We prove no-dimensional variants of the Tverberg theorem and its colorful counterpart that allow efficient algorithms to find the partition. Our proofs are inspired by Sarkaria’s proof [15] and the averaging technique by Adiprasito, Bárány, and Mustafa [1]. For the colorful version, we additionally make use of ideas from Soberón’s proof [16].

More precisely, our results are as follows:

  • Sarkaria’s method uses vectors in to lift the points in the Tverberg instance to a colorful Carathéodory instance. We refine this method to vectors that are defined with the help of a given graph. The choice of this graph is important in proving good bounds for our results and in the algorithm. We believe that this generalization is of an independent interest and may prove useful in other scenarios that make use of the tensor product construction.

  • We prove an efficient no-dimensional Tverberg result:

    Theorem 1.1 (efficient no-dimensional Tverberg theorem).

    Let be a set of points, and let be an integer.

    • For any choice of positive integers that satisfy , there is a partition of with , and a -dimensional ball of radius

      such that intersects the convex hull of each .

    • The bound is better for the case and . There exists a partition of with and a -dimensional ball of radius

      that intersects the convex hull of each .

    In either case, we can compute the partition in deterministic time

  • and a colorful counterpart (for a simple example, see Figure 1):

    Theorem 1.2 (efficient no-dimensional Colorful Tverberg).

    Let be point sets, each of size , with being a positive integer, so that the total number of points is . Then, there are pairwise-disjoint colorful sets and a -dimensional ball of radius

    that intersects the convex hull of each . We can find the ’s in deterministic time

Figure 1: Left: a point set on three colors and four points of each color. Right: a colorful partition with a ball containing the centroids (squares) of the sets of the partition.

The colorful result is similar in spirit to the regular Tverberg result from Section 2, but for computational considerations, it currently does not make sense to use the colorful version to solve the regular Tverberg problem.

Compared to the results of Adiprasito et al. [1], our radius bounds are slightly worse. More precisely, they show that both in the colorful and the non-colorful case, there is a ball of radius that intersects the convex hulls of the sets of the partition. They also show this bound is close to optimal. In contrast, our result is off by a factor of , but the proof technique of Adiprasito et al. [1] gives only a brute-force algorithm, which is not efficient. Our approach, however, gives almost linear time algorithms for both cases, with a linear dependence on the dimension.

Adiprasito et al. first prove the colorful no-dimensional Tverberg theorem using an averaging argument over an exponential number of possible partitions. Then, they specialize their result for the regular case, obtaining a bound that is asymptotically optimal. Unfortunately, it is not clear how to derandomize the averaging argument efficiently. To get around this, we follow an alternative approach towards both versions of the Tverberg theorem. Instead of a direct averaging argument, we use a reduction to the Colorful Carathéodory theorem that is inspired by Sarkaria’s proof, with some additional twists. We will see that this reduction also works in the no-dimensional setting, i.e., by a reduction to the no-dimensional Colorful Carathéodory theorem of Adiprasito et al., we obtain a no-dimensional Tverberg theorem, with slightly weaker radius bounds, as stated above. This approach has the advantage that their Colorful Carathéodory theorem is based on an averaging argument that permits an efficient derandomization using the method of conditional expectations [2]. In fact, we will see that the special structure of our Colorful Carathéodory instance allows for a very fast evaluation of the conditional expectations, as we fix the next part of the solution. This results in an algorithm whose running time is instead of , as given by a naive application of the method. With a few interesting modifications, this idea also works in the colorful setting.

Outline of the paper.

We begin by describing our extension of Sarkaria’s technique in Section 2 and then use it in combination with a result from Section 3 to prove the no-dimensional Tverberg result. In Section 3, we expand upon the details of an averaging argument that is useful for the Tverberg result. Section 4 is devoted to describing an algorithm to compute the Tverberg partition. In Section 5 we give a corresponding result for the colorful Tverberg setting and describe an algorithm to compute the required partition. We conclude in Section 6 with some observations and open questions.

2 Tensor product and no-dimensional Tverberg theorem

In this section, we prove a no-dimensional Tverberg result. Let denote the diameter of any point set in dimensions. Let be our given set of points in dimensions. We assume for simplicity that the centroid of , that we denote by , coincides with the origin , i.e., . For ease of presentation, we denote the origin by in all dimensions, as long as there is no danger of confusion. Also, we use to denote the usual scalar product between two vectors in the appropriate dimension.

Tensor product.

Let and be any two vectors in and dimensions, respectively. The tensor product is the operation that takes and to the -dimensional vector

Straightforward calculations show that for any vectors , the operator satisfies:

  1. [(i)]

  2. ,

  3. , and

  4. .

By (iii), the -norm of the tensor product is exactly . For any set of vectors in and any -dimensional vector , we denote by the set of tensor products . Throughout this paper, all distances will be in the -norm.

A set of lifting vectors.

We generalize the tensor construction that was used by Sarkaria to prove the Tverberg theorem [15]. For this, we provide a way to construct a set of vectors that we use to create tensor products. The motivation behind the precise choice of these vectors will be explained a little later in this section. Let be an (undirected) simple, connected graph of nodes and let

  • denote the number of edges in ,

  • denote the maximum degree of any node in , and

  • denote the diameter of , i.e., the maximum length of a shortest path between a pair of vertices in .

We orient the edges of in an arbitrary manner to obtain a directed graph. We use this directed version of to define a set of vectors in dimensions. This is done as follows: each vector corresponds to a unique node of . Each coordinate position of the vectors corresponds to a unique edge of . If is a directed edge of , then contains a and contains a in the corresponding coordinate position. That means, the vectors are in . Also, . It can be verified that this is the unique linear dependence (up to scaling) between the vectors for any choice of edge orientations. This means that the rank of the matrix with the s as the rows is . The squared norm is the degree of , for each vertex . For , the dot product is if is an edge in , and otherwise.

For any set of vectors, each of the same dimension, we note that property (iii) of the tensor product leads to the following relation:

(1)

where is the set of edges of .

As an example, such a set of vectors can be formed by taking as a balanced binary tree with nodes, and orienting the edges away from the root. Let correspond to the root. A simple instance of the vectors is shown below:

The vectors in the figure above can be represented as the matrix

where the -th row of the matrix corresponds to vector . As , each vector is in . The norm is one of , , or , depending on whether is the root, an internal node with two children, or a leaf, respectively. The height of is and the maximum degree is .

Lifting the point set.

Let . Our goal is to find a (relaxed) Tverberg partition of into sets. For this, we first pick a graph with vertices, as in the previous paragraph, and we derive a set of lifting vectors from . Then, we lift each point of to a set of vectors in dimensions, by taking tensor products with the vectors . More precisely, for and , let

For , we let be the lifted points obtained from . We have,

By the bi-linear properties of the tensor product, we have

so the centroid coincides with the origin, for .

The next lemma contains the technical core of our argument. It shows how to use the lifted point sets to derive a useful partition of into subsets of prescribed sizes. We defer its proof to Section 3.

Lemma 2.1.

Let be a set of points in and let denote the point sets obtained by lifting each using the vectors .

For any choice of positive integers that satisfy , there is a partition of with such that the centroid of the set of lifted points (which is a traversal of ) has distance less than

from the origin .

The bound is better for the case and . There exists a partition of with such that the centroid of has distance less than

from the origin .

Using Lemma 2.1, we show that there is a ball of bounded radius that intersects the convex hull of each . Let be positive real numbers. The centroid of can be written as

where denotes the centroid of , for .

Using Equation (1),

(2)

Let . Then

so the centroid of coincides with the origin. Using and Equation (2),

We bound the distance from to every other . For each , we associate to the node in . Then the shortest path from to in has length at most . Let that path be denoted by . Using triangle inequality and the Cauchy-Schwarz inequality,

(3)

Therefore, the ball of radius centered at covers the set . That means, the ball covers the convex hull of and in particular contains the origin. Using triangle inequality, the ball of radius centered at the origin contains . Then the norm of each is at most which implies that the norm of each is at most . Therefore, the ball of radius

centered at contains the set . Substituting the value of from Lemma 2.1, the ball of radius

centered at covers the set .

Optimizing the choice of .

The radius of the ball has a term that depends on the choice of . For a path graph this term has value and for a star graph this is . If is a balanced -ary tree, then the Cauchy-Schwarz inequality in Equation (3) can be modified to replace by the height of the tree. Then the term is which is minimized for . The radius bound for this choice of is

as claimed in Theorem 1.1.

Balanced partition.

For the case and , we give a better bound for the radius of the ball containing the centroids . In this case we have . Then Equation (2) is

Since , we get

(4)

Similar to the general case, we bound the distance from to any other centroid . For each , we associate to the node in . There is a path of length at most from to any other node. Using the Cauchy-Schwarz inequality and substituting the value of , we see that

(5)

Therefore, a ball of radius centered at contains the set . The factor is minimized when is a star graph, that is, a tree with one root and children. Then the ball containing has radius

as claimed in Theorem 1.1.

As balanced as possible.

When does not divide , but we still want a balanced partition, we take any subset of points of and get a balanced Tverberg partition on the subset. Then we add the removed points one by one to the sets of the partition, adding at most one point to each set.

As shown above, there is a ball of radius less than that intersects the convex hull of each set in the partition. Noting that

a ball of radius less than intersects the convex hull of each set of the partition.

3 Existence of a desired partition

This section is dedicated to the proof of Lemma 2.1. Like Adiprasito et al. [1], we use an averaging argument to obtain the result. More precisely, we bound the average norm of the centroid of the lifted points over all partitions of of the form , for which the sets in the partition have sizes respectively, with .

Each such partition can be considered as a traversal of the lifted point sets . Thus, consider any traversal of , where , for . The centroid of is . We bound the expectation , over all possible traversals . The expectation can be written as

We next find the coefficient of each term of the form and in the expectation. Using the multinomial coefficient, the total number of traversals is

Furthermore, for any lifted point , the number of traversals with is

So the coefficient of is Similarly, for any pair of points , there are two cases in which they appear in the same traversal:

  • : the number of traversals is . The coefficient of in the expectation is hence .

  • : the number of traversals is calculated to be . The coefficient of in the expectation is .

Substituting the coefficients, we bound the expectation as

We bound the value of each of the three terms individually to get an upper bound on the value of the expression. The first term can be bounded as

where we have made use of the fact that (see [1, Lemma  7.1]). The second term can be re-written as

where we have again made use of [1, Lemma 7.1] to bound the term

The second term is non-positive and therefore can be removed since the total expectation is always non-negative. The third term is