  Let $P$ be a $d$-dimensional $n$-point set. A partition $T$ of $P$ is called
a Tverberg partition if the convex hulls of all sets in $T$ intersect in at
least one point. We say $T$ is $t$-tolerant if it remains a Tverberg partition
after deleting any $t$ points from $P$. Sober\'{o}n and Strausz proved that
there is always a $t$-tolerant Tverberg partition with $\lceil n / (d+1)(t+1)
\rceil$ sets. However, so far no nontrivial algorithms for computing or
approximating such partitions have been presented.
  For $d \leq 2$, we show that the Sober\'{o}n-Strausz bound can be improved,
and we show how the corresponding partitions can be found in polynomial time.
For $d \geq 3$, we give the first polynomial-time approximation algorithm by
presenting a reduction to the Tverberg problem with no tolerance. Finally, we
show that it is coNP-complete to determine whether a given Tverberg partition
is t-tolerant.
