Let P be a d-dimensional n-point set. A Tverberg-partition of P is a partition of P into r sets P_1, ..., P_r such that the convex hulls conv(P_1), ..., conv(P_r) have non-empty intersection. A point in the intersection of the conv(P_i)'s 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)^3 in time d^{O(log d)} n. 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 (2010).Comment: 14 pages, 2 figures. A preliminary version appeared in SoCG 201

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].

We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension d:i) Given n points in R d , compute their minimum enclosing cylinder.ii) Given two n-point sets in R d , decide whether they can be separated by two hyperplanes.iii) Given a system of n linear inequalities with d variables, find a maximum-size feasible subsystem.We show that (the decision versions of) all these problems are W[1]-hard when parameterized by the dimension d. Our reductions also give a n Ω(d) -time lower bound (under the Exponential Time Hypothesis).

For q ≥ 27 we determine the independence number α( ) of the Kneser graph on plane-solid flags in PG(6, q). More precisely we describe all maximal independent sets of size at least q 11 and show that every other maximal example has cardinality at most a constant times q 10 .

We study the following general stabbing problem from a parameterized complexity point of view: Given a set $\mathcal S$ of $n$ translates of an object in $\Rd$, find a set of $k$ lines with the property that every object in $\mathcal S$ is ''stabbed'' (intersected) by at least one line. We show that when $S$ consists of axis-parallel unit squares in $\Rtwo$ the (decision) problem of stabbing $S$ with axis-parallel lines is W[1]-hard with respect to $k$ (and thus, not fixed-parameter tractable unless FPT=W[1]) while it becomes fixed-parameter tractable when the squares are disjoint. We also show that the problem of stabbing a set of disjoint unit squares in $\Rtwo$ with lines of arbitrary directions is W[1]--hard with respect to $k$. Several generalizations to other types of objects and lines with arbitrary directions are also presented. Finally, we show that deciding whether a set of unit balls in $\Rd$ can be stabbed by one line is W[1]--hard with respect to the dimension $d$.Comment: Based on the MSc. Thesis of Daniel Werner, Free University Berlin, Berlin, German