Tverberg's theorem states that for any $k \ge 2$ and any set $P \subset \mathbb{R}^d$ of at least $(d + 1)(k - 1) + 1$ 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 $\text{CLS} = \text{PPAD} \cap \text{PLS}$, 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, Barany, and Mustafa 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 $P \subset \mathbb{R}^d$ and any $k \in \{2, \dots, n\}$ in $O(nd \lceil{\log k}\rceil)$ time a $k$-partition of $P$ such that there is a ball of radius $O\left((k/\sqrt{n})\mathrm{diam(P)}\right)$ 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 to reduce the Tverberg problem to the Colorful Caratheodory problem (in the simplified tensor product interpretation of Barany 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.

翻译：Tverberg定理指出：对于任意 $k \ge 2$ 及 $d$ 维空间中至少包含 $(d+1)(k-1)+1$ 个点的点集 $P \subset \mathbb{R}^d$，总可将 $P$ 划分为 $k$ 个子集，使得这些子集的凸包存在非空交集。寻找该划分的关联搜索问题属于复杂度类 $\text{CLS} = \text{PPAD} \cap \text{PLS}$，但目前尚未发现其困难性结果。在彩色Tverberg定理中，点集 $P$ 中的点具有颜色，且在特定条件下，$P$ 可被划分为若干彩色集合——每个颜色恰好出现一次，且这些彩色集合的凸包存在交集。截至目前，该关联搜索问题的复杂度尚未解决。近期，Adiprasito、Barany 与 Mustafa 提出了一种无维数Tverberg定理，其中凸包可近似相交。该定理放宽了对 $P$ 基数的要求。其论证是构造性的，但未能给出多项式时间算法。我们提出一个确定性算法：对于任意 $n$ 点集 $P \subset \mathbb{R}^d$ 及任意 $k \in \{2, \dots, n\}$，可在 $O(nd \lceil{\log k}\rceil)$ 时间内找到一个 $k$-划分，使得存在一个半径为 $O\left((k/\sqrt{n})\mathrm{diam(P)}\right)$ 的球，该球与每个子集凸包相交。鉴于该问题目前尚无法在多项式时间内精确求解，我们的结果提供了一种极为高效且简洁的近似新概念。主要贡献在于推广了Sarkaria方法，将Tverberg问题约化为彩色Carathéodory问题（基于Barany与Onn的简化张量积解释），并对其进行算法化实现。事实证明，该方法不仅为无维数Tverberg定理提供了替代性算法证明，还能推广至该问题的彩色变体等其他场景。