Given two simple polygons P and Q we define the weight of a bridge (p,q), with p ∈ ρ(P) and q ∈ ρ(Q), where ρ() denotes the compact region enclosed by the boundary of the polygon, between the two polygons as gd(p,P) + d(p,q) + gd(q,Q), where d(p,q) is the Euclidean distance between the points p and q, and gd(x,X) is the geodesic distance between x and its geodesic furthest neighbor on X. Our problem differs from another version of the problem where the additional restriction of requiring the endpoints of the bridge to be mutually visible was imposed. We show that an optimal bridge always exists such that the endpoints of the bridge lie on the boundaries of the two polygons. Using this critical property, we present an algorithm to find an optimal bridge (of minimum weight) in O(n2 log n) time. We present a polynomial time approximation scheme that for any ∊ > 0 generates a bridge with objective function within a factor of 1 + ∊ of the optimal value in O(kn log kn) time, where [Formula: see text]. An improved polynomial time approximation scheme and algorithms for generalized versions of our problems are also discussed.
Polynomial Time Approximation Scheme for the Minimum-weight k-Size Cycle Cover Problem in Euclidean space of an arbitrary fixed dimension
