A $(5/3+ε)$-Approximation for Tricolored Non-crossing Euclidean TSP

TL;DR

This work extends Arora’s Euclidean TSP framework to the 3-colored, non-crossing setting ($3$-ETSP) by developing a tri-color patching mechanism and a portal-based dissection scheme. When patching three tours is feasible, the authors obtain a $(5/3+ ext{ε})$-approximation; otherwise they reduce to a weighted two-tour problem and solve it via a dynamic-programming DP over a portal grid. A key component is a structure theorem: for any ε>0, either a portal-respecting $(1+ε)$-approximate three-tour solution exists, or a portal-respecting two-tour presolution with a $(5/3+ε)$-approximation exists, enabling a single DP-based pipeline to compute a near-optimal portal-respecting solution. Perturbation and back-perturbation steps ensure the method applies to general instances by mapping to a fine grid and recovering a near-optimal solution to the original input, with running time bounded by $(n/ ext{ε})^{O(1/ ext{ε})}$. Overall, the paper delivers a near-optimal algorithm for a challenging multi-tour, non-crossing Euclidean setting and strengthens the understanding of tri-color geometric TSP via portal-based DP techniques.

Abstract

In the Tricolored Euclidean Traveling Salesperson problem, we are given~$k=3$ sets of points in the plane and are looking for disjoint tours, each covering one of the sets. Arora (1998) famously gave a PTAS based on ``patching'' for the case $k=1$ and, recently, Dross et al.~(2023) generalized this result to~$k=2$. Our contribution is a $(5/3+ε)$-approximation algorithm for~$k=3$ that further generalizes Arora's approach. It is believed that patching is generally no longer possible for more than two tours. We circumvent this issue by either applying a conditional patching scheme for three tours or using an alternative approach based on a weighted solution for $k=2$.

Figures (12)

• Figure 1: An instance of $3\text{-}\mathrm{ETSP}\xspace$ together with two possible solutions. An optimum solution does not exist: The curves can get arbitrarily close but must not touch. Observe that, in the middle subfigure, the red and green tour are not $\delta$-close for any $\delta>0$, as the blue tour lies in between.
• Figure 2: A modified example from bereg that is presumably non-patchable.
• Figure 3: On the left, we have a single curve $\pi_{\mathrm{R}\mathrm{G}}$ visiting all red and green points. On the right, we have replaced $\pi_{\mathrm{R}\mathrm{G}}$ by two parametrized disjoint curves $\pi_{\mathrm{R}}(\lambda)$ and $\pi_{\mathrm{G}}(\lambda)$ with Fréchet-distance at most $\lambda$ to $\pi_{\mathrm{R}\mathrm{G}}$, visiting the terminals of the corresponding color. In particular, the Fréchet-distance between $\pi_{\mathrm{R}}$ and $\pi_{\mathrm{G}}$ is at most $2 \lambda$.
• Figure 4: On the left, we are given a solution to $3\text{-}\mathrm{ETSP}\xspace$ where the red and green tour are $\delta$-close. On the right, we see how the solution can be transformed into an induced two-tour solution.
• Figure 5: The figure on the left illustrates the dissection $D(\boldsymbol{a})$ with $L=4, \boldsymbol{a}=(1,0)$. The dashed lines denote $\partial [0,L]^2$. The three pink lines are examples of boundaries of levels one, two, and three. The levels of all horizontal grid lines are indicated. We have placed 4 portals on every boundary, represented as circles. For better overview, we have drawn only one portal at endpoints of boundaries. Note that the endpoint of a boundary is actually contained in up to four portals. This is illustrated on the right hand side, where the exact portal placement of the marked orange area is given.

Theorems & Definitions (28)

• Theorem 1
• Lemma 2
• Lemma 3
• Theorem 4
• Theorem 5
• Lemma 6
• Lemma 8
• Lemma 9: Arora arora
• Lemma 10: Dross et al. dross
• Lemma 10: Tricolored Patching