A Polynomial Time Approximation Scheme for the Problem of Interconnecting Highways

The objective of the Interconnecting Highways problem is to construct roads of minimum total length to interconnect n given highways under the constraint that the roads can intersect each highway only at one point in a designated interval which is a line segment. We present a polynomial time approximation scheme for this problem by applying Arora’s framework in [2]. For every fixed c > 1 and given any n line segments in the plane, a randomized version of the scheme finds a 1+ c -approximation to the optimal cost in O(nO(c)log(n)) time.