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The associated search problem of finding the partition lies in the complexity class $$\\text {CLS} = \\text {PPAD} \\cap \\text {PLS}$$<\/jats:tex-math>\n \n CLS<\/mml:mtext>\n =<\/mml:mo>\n PPAD<\/mml:mtext>\n \u2229<\/mml:mo>\n PLS<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, but no hardness results are known. In the colorful<\/jats:italic> Tverberg theorem, the points in P<\/jats:italic> have colors, and under certain conditions, P<\/jats:italic>\u00a0can be partitioned into colorful<\/jats:italic> 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, B\u00e1r\u00e1ny, and Mustafa (SODA 2019) gave a no-dimensional<\/jats:italic> Tverberg theorem, in which the convex hulls may intersect in an approximate<\/jats:italic> fashion. This relaxes the requirement on the cardinality of\u00a0P<\/jats:italic>. The argument is constructive, but does not result in a polynomial-time algorithm. We present a deterministic algorithm that finds for any n<\/jats:italic>-point set $$P\\subset {\\mathbb {R}}^d$$<\/jats:tex-math>\n \n P<\/mml:mi>\n \u2282<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and any $$k\\in \\{2,\\dots ,n\\}$$<\/jats:tex-math>\n \n k<\/mml:mi>\n \u2208<\/mml:mo>\n {<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n n<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$O(nd\\lceil {\\log k}\\rceil )$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n d<\/mml:mi>\n \u2308<\/mml:mo>\n \n log<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n \u2309<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> time a k<\/jats:italic>-partition of P<\/jats:italic> such that there is a ball of radius $$O((k\/\\sqrt{n}){\\text {diam}}(P))$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \n (<\/mml:mo>\n k<\/mml:mi>\n \/<\/mml:mo>\n \n n<\/mml:mi>\n <\/mml:msqrt>\n )<\/mml:mo>\n <\/mml:mrow>\n diam<\/mml:mtext>\n \n (<\/mml:mo>\n P<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> 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\u2019s method (Israel Journal Math., 1992) to reduce the Tverberg problem to the colorful Carath\u00e9odory problem (in the simplified tensor product interpretation of B\u00e1r\u00e1ny 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.<\/jats:p>","DOI":"10.1007\/s00454-022-00380-1","type":"journal-article","created":{"date-parts":[[2022,4,12]],"date-time":"2022-04-12T14:04:09Z","timestamp":1649772249000},"page":"964-996","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["No-Dimensional Tverberg Theorems and Algorithms"],"prefix":"10.1007","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9225-0829","authenticated-orcid":false,"given":"Aruni","family":"Choudhary","sequence":"first","affiliation":[]},{"given":"Wolfgang","family":"Mulzer","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,12]]},"reference":[{"key":"380_CR1","doi-asserted-by":"crossref","unstructured":"Adiprasito, K., B\u00e1r\u00e1ny, I., Mustafa, N.H.: Theorems of Carath\u00e9odory, Helly, and Tverberg without dimension. 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