Fixed-parameter tractability and lower bounds for stabbing problems

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 $d$.