Gap-ETH-Tight Approximation Schemes for Red-Green-Blue Separation and Bicolored Noncrossing Euclidean Travelling Salesman Tours

TL;DR

This paper presents Gap-ETH-tight approximation schemes for noncrossing point connection problems, including the two-colored TSP and red-blue-green separation, on Euclidean plane and planar graphs, advancing previous approximation algorithms.

Contribution

It introduces new Gap-ETH-tight EPTAS for multi-colored noncrossing connection problems and a novel patching procedure, improving upon prior work.

Findings

EPTAS for two-colored TSP and red-green-blue separation on Euclidean plane

PTAS for two-colored TSP in unweighted planar graphs

NP-hardness results for noncrossing path and spanning tree problems

Abstract

In this paper, we study problems of connecting classes of points via noncrossing structures. Given a set of colored terminal points, we want to find a graph for each color that connects all terminals of its color with the restriction that no two graphs cross each other. We consider these problems both on the Euclidean plane and in planar graphs. On the algorithmic side, we give a Gap-ETH-tight EPTAS for the two-colored traveling salesman problem as well as for the red-blue-green separation problem (in which we want to separate terminals of three colors with two noncrossing polygons of minimum length), both on the Euclidean plane. This improves the work of Arora and Chang (ICALP 2003) who gave a slower PTAS for the simpler red-blue separation problem. For the case of unweighted plane graphs, we also show a PTAS for the two-colored traveling salesman problem. All these results are based on our new patching procedure that might be of independent interest. On the negative side, we show that the problem of connecting terminal pairs with noncrossing paths is NP-hard on the Euclidean plane, and that the problem of finding two noncrossing spanning trees is NP-hard in plane graphs.