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 (Arora, 1998; also available from http:www.cs.princeton.edu/~arora). For every fixed c > 1 and given any n line segments in the plane, a randomized version of the scheme finds a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left( {1 + \frac{1}{c}} \right)$$ \end{document}-approximation to the optimal cost in O(nO(c)log(n) time.