Colored Non-Crossing Euclidean Steiner Forest

TL;DR

This paper introduces approximation algorithms for the colored non-crossing Euclidean Steiner forest problem, providing a PTAS for two colors and improved ratios for three and general k, advancing solutions for non-crossing geometric network design.

Contribution

It presents the first PTAS for two-colored instances and improved approximation algorithms for three and general k, addressing a complex geometric network problem.

Findings

PTAS achieved for k=2 case.

Approximation ratio of 5/3+ε for k=3.

New algorithms with O(√n log k) and k+ε ratios for general k.

Abstract

Given a set of $k$-colored points in the plane, we consider the problem of finding $k$ trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For $k=1$, this is the well-known Euclidean Steiner tree problem. For general $k$, a $k\rho$-approximation algorithm is known, where $\rho \le 1.21$ is the Steiner ratio. We present a PTAS for $k=2$, a $(5/3+\varepsilon)$-approximation algorithm for $k=3$, and two approximation algorithms for general~$k$, with ratios $O(\sqrt n \log k)$ and $k+\varepsilon$.