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I'm interested in the parameterized complexity of what I'll call the d-Dimensional Hitting Set problem: given a range space (i.e. a set system / hypergraph) S = (X,R) having VC-dimension at most d and a positive integer k, does X contain a subset of size k that hits every range in R? The parameterized version of the problem is parameterized by k.

For what values of d is the d-Dimensional Hitting Set problem

  • in FPT?
  • in W[1]?
  • W[1]-hard?
  • W[2]-hard?

What I know can be summarized as follows:

  • 1-Dimensional Hitting Set is in P and is therefore in FPT. If S has dimension 1 then it is not difficult to show that either there is a hitting set of size 2 or the incidence matrix of S is totally balanced. In either case we can find a minimum hitting set in polynomial time.

  • 4-Dimensional Hitting Set is W[1]-hard. Dom, Fellows, and Rosamond [PDF] proved W[1]-hardness for the problem of stabbing axis-parallel rectangles in R^2 with axis-parallel lines. This can be formulated as Hitting Set in a range space of VC-dimension 4.

  • If no restriction is placed on d then we have the standard Hitting Set problem which is W[2]-complete and NP-complete.

  • Langerman and Morin [citeseer link] give FPT algorithms for Set Cover in restricted dimension, though their bounded dimensionality model is not the same as the model defined by bounded VC-dimension. Their model does not seem to include, for example, the problem of hitting halfspaces with points, though the prototype problem for their model is equivalent to hitting hyperplanes with points.

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    $\begingroup$ Great question! $\endgroup$ – András Salamon Aug 17 '10 at 17:58
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I think this problem is too hard. We do not know the answer to much easier problems in this family. For example, given a set of n points in the plane, and a set of (say n) unit disks, decide if there is a cover of the points by k of the unit disks. There is an easy n^O(k) time algorithm for this, and I would not be surprised if using known insights one can do n^O(sqrt{k}) (but even that is not obvious), but doing f(k)*n^{O(1)} is open, and in fact would be quite interesting. A (1+eps) approximation follows from the work of Mustafa and Ray http://portal.acm.org/citation.cfm?id=1542362.1542367.

BTW, for the continuous version where any unit disk is allowed, one can solve the problem in n^{O(k)} time. A PTAS in this case is also pretty easy using shifted grids.

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We address this question in a new preprint: http://arxiv.org/abs/1512.00481

Hitting Set in hypergraphs of low VC-dimension (Karl Bringmann, László Kozma, Shay Moran, N.S. Narayanaswamy).

It turns out that Hitting Set is W[1]-hard already when the VC-dimension is equal to 2.

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The paper Dániel Marx: Efficient Approximation Schemes for Geometric Problems?. ESA 2005: 448-459 is quite relevant.

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