Algorithms for Tolerated Tverberg Partitions

Robust Tverberg and Colourful Carathéodory Results via Random Choice

“…The advantage of the result above is that generating the partition is trivial, taking time N. The problem of finding Tverberg partitions, in both its deterministic and non-deterministic versions, is interesting. See, for instance, [15,26,28,31].…”

“…The Soberón-Strausz bound is known not to be optimal, as shown by Mulzer and Stein for d = 1 and in some instances for d = 2 [MS13]. Recently, this was vastly improved by García-Colín, Raggi and Roldán-Pensado [GCRRP16], who showed that for fixed r, d we have N (t, d, r) = rt + o(t).…”

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…A tolerant Tverberg theorem, due to Soberón and Strausz [357], asserts that any set of (t+1)(r−1)(d+1)+1 points can be partitioned into r parts such that, after deletion of any t points, what remains is a Tverberg partition. This bound was improved to r(t + 2) − 1 for d = 1 and to 2r(t + 2) − 1 for d = 2 [284] (bound for d = 1 is tight). Recently there have been two significant improvements, García-Colín et al [179] gave an asymptotically tight bound for the tolerant Tverberg theorem when the dimension and the size of the partition are fixed.…”

Tolerance for colorful Tverberg partitions

“…There are improved bounds on t for small dimension or small number of parts [SS12,MS13,BH20]. The bounds on the o(N ) term have been improved to be polynomial in terms of N, r, d, and for r, d fixed it can be replaced by O( N ln(N )) using the probabilistic method [Sob18,Sob19].…”