Approximation algorithms for hitting objects with straight lines

In the hitting set problem one is given m subsets of a nite set N and one has to nd an X N of minimum cardinality that hits intersects all of them The problem is NP hard It is not known whether there exists a polynomial time approximation algorithm for the hitting set problem with a nite performance ratio Special cases of the hitting set problem are described for which nite performance ratios are guaranteed These problems arise in a geometric setting We consider special cases of the following problem Given n compact subsets of R nd a set of straight lines of minimum cardinality so that each of the given subsets is hit by at least one line The algorithms are based on several techniques of representing objects by points not necessarily points on the objects and solving in some cases only approximately the problem of hitting the representative points Finite performance ratios are obtained when the dimension the number of types of sets to be hit and the number of directions of the hitting lines are bounded