Quasilinear Approximation Scheme for Steiner Multi Cycle in the Euclidean plane.

We propose a randomized approximation scheme for the Euclidean Steiner Multi Cycle problem which runs in quasilinear time. In this problem, we are given a set of n pairs of points (terminals) T={{ti,ti′}}i=1n in the Euclidean plane, and the objective is to find a collection of cycles of minimum cost such that ti and ti′ belong to a same cycle, for each i ∈ {1, ..., n}. This problem extends the Steiner Cycle problem in the same way the Steiner Forest extends the Steiner Tree problem. Additionally, it has applications on routing problems with pickup and delivery locations.