# Approximation Algorithms for Piercing Special Families of Hippodromes: An Extended Abstract

## Abstract

Polynomial time approximation algorithms are proposed with constant approximation factors for a problem of computing the smallest cardinality set of identical disks whose union intersects each segment from a given set *E* of *n* straight line segments on the plane. This problem has important applications in operations research, namely in wireless and road network analysis. It is equivalent to finding the least cardinality piercing (or hitting) set for the corresponding family of *n* Euclidean *r*-neighborhoods of straight line segments of *E* on the plane, which are called *r*-hippodromes in the literature. When the number of distinct orientations is upper bounded by *k* of segments from *E*, a simple \(O(n\log n)\)-time 4*k*-approximate algorithm is known for this problem. Besides, when *E* contains arbitrary straight line segments, overlapping at most at their endpoints, \(O(n^4\log n)\)-time 100-approximate algorithm is given recently. In the present paper, simple approximation algorithms are proposed with small approximation factors for *E*, being edge set of some special plane graphs of interest in road network applications; here the number of distinct orientations of straight line segments from *E* can be arbitrarily large. More precisely, \(O(n^2)\)-time approximation algorithms are constructed for edge sets of either Gabriel or relative neighborhood graphs or of Euclidean minimum spanning trees with factors of 14, 12 and 10 respectively. These algorithms are much faster, more accurate and conceptually much simpler than the aforementioned 100-approximate algorithm for the general case of the problem on edge sets of arbitrary plane graphs.

## Keywords

Operations research Computational geometry Approximation algorithms Geometric piercing problem Straight line segment Hippodrome Proximity graph## References

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