Approximating Tverberg points in linear time for any fixed dimension

Abstract

Let P ⊆ R d be a d-dimensional n-point set. A Tverberg partition of P is a partition of P into r sets P1, . . . , Pr such that the convex hulls conv(P1), . . . , conv(Pr) have non-empty intersection. A point in r i=1 conv(Pi) is called a Tverberg point of depth r for P . A classic result by Tverberg implies that there always exists a Tverberg partition of size n/(d + 1) , but it is not known how to find such a partition in polynomial time. Therefore, approximate solutions are of interest.We describe a deterministic algorithm that finds a Tverberg partition of size n/4(d + 1)This means that for every fixed dimension we can compute an approximate Tverberg point (and hence also an approximate centerpoint) in linear time. Our algorithm is obtained by combining a novel lifting approach with a recent result by Miller and Sheehy [10].

A resilient convex combination for consensus-based distributed algorithms

“…Although the Tverberg Theorem provides a sufficient condition for the existence of a partition leading to Tverberg point, the theorem does not provide an algorithm for finding the partition for achieving Tverberg points, apart from enumerating all possible partitions and checking the intersection of convex hulls for each partition. As mentioned in [1,18], except for some specific values of n, the computational complexity of achieving Tverberg points grows exponentially with the dimension n. In contrast, it has been shown that the resilient convex combination proposed in (17) can be computed by solving a standard quadratic programming problem, whose computational complexity is polynomial in n. • Second, the existence of an non-empty Tverberg point set T requires that m ≥ κ(n + 1) + 1, while one has R = ∅ ifĀ = ∅ or m ≥ κ(n + 1) + 1.…”

“…Then there must exist a partition of A into κ + 1 disjoint subsets B 1 , · · · , B κ+1 such that While results in [15,22,28,30] are elegant, one major concern of applying Tverberg points lies in the requirement of high computational complexity. As mentioned in [1,18], except for some specific values of n, the computational complexity of calculating Tverberg points grows exponentially with the dimension n. In the following, we will develop a low-complexity algorithm for achieving resilient convex combinations based on the intersection of convex hulls.…”

A New Approach for Distributed Hypothesis Testing with Extensions to Byzantine-Resilience

“…While this can indeed be achieved efficiently for scalar consensus problems, for problems requiring consensus on vectors (like the belief vectors in our setting), such an approach typically requires the computation of sets known as Tverberg partitions. However, there is no known algorithm that can compute an exact Tverberg partition in polynomial time for a general ddimensional finite point set [39]. Consequently, since the filtering approach developed in [14] requires each regular agent to compute a Tverberg partition at every iteration, the resulting computations are forbiddingly high.…”

“…where a i,t (θ) = t − c i,t (θ), η i,t (θ ) is as defined in (39), and x(τ ), τ ∈ {a i,t (θ) + 1, . .…”

Multidimensional agreement in Byzantine systems

“…The deterministic procedure in the algorithm could therefore return a Tverberg point. For arbitrary d, no algorithm to compute a Tverberg point of an arbitrary multiset is currently known with polynomial complexity [2,18,19]. However, in some restricted cases, efficient algorithms are known (e.g., [15]).…”