NP-hardness and a PTAS for the Euclidean Steiner Line Problem

TL;DR

This work studies the Euclidean Steiner Line Problem (ESL) and its fixed-line variant (ESfL), where a zero-cost line $\gamma$ must be incorporated into a minimum-length Steiner tree. It resolves open questions by proving NP-hardness for both ESL and ESfL via reductions from a special case of the EST (PALIMEST), and by establishing a PTAS for both problems through a reduction to the classic EST. The PTAS leverages a discretization of $\gamma$ into line points, an EST-PTAS, and a post-processing FillHoles routine to preserve line-structure, achieving a randomized runtime of $O\big(2^{O(1/\varepsilon)}\cdot\frac{n}{\varepsilon} + \mathrm{poly}(1/\varepsilon)\cdot\frac{n}{\varepsilon}\log\frac{n}{\varepsilon} + \big(\frac{n}{\varepsilon}\big)^3\big)$. The ESL case follows by applying the ESfL-PTAS to all feasible placements of $\gamma$ through two terminals, yielding a practical near-optimal framework for line-constrained Euclidean Steiner networks. These results deepen our understanding of line-augmented Steiner problems and provide near-optimal algorithms with provable guarantees for geometric networks.

Abstract

The Euclidean Steiner Tree Problem (EST) seeks a minimum-cost tree interconnecting a given set of terminal points in the Euclidean plane, allowing the use of additional intersection points. In this paper, we consider two variants that include an additional straight line $γ$ with zero cost, which must be incorporated into the tree. In the Euclidean Steiner fixed Line Problem (ESfL), this line is given as input and can be treated as a terminal. In contrast, the Euclidean Steiner Line Problem (ESL) requires determining the optimal location of $γ$. Despite recent advances, including heuristics and a 1.214-approximation algorithm for both problems, a formal proof of NP-hardness has remained open. In this work, we close this gap by proving that both the ESL and ESfL are NP-hard. Additionally, we prove that both problems admit a polynomial-time approximation scheme (PTAS), by demonstrating that approximation algorithms for the EST can be adapted to the ESL and ESfL with appropriate modifications. Specifically, we show ESfL$\leq_{\text{PTAS}}$EST and ESL$\leq_{\text{PTAS}}$EST, i.e., provide a PTAS reduction to the EST.

Figures (6)

• Figure 1: Illustration of a wedge $W$ (shaded in gray) that does not contain any terminals and has no intersection with $\gamma$. Let $s$ be the Steiner point in $W$ with the largest $y$-coordinate. As at least one edge of $s$ faces upwards and cannot leave the wedge, we arrive at a contradiction, proving that in any optimal solution $W$ does not contain any Steiner points.
• Figure 6: The line between the leftmost and rightmost terminals is fully covered by pyramids. Therefore, the width of the instance is bounded by the sum of the widths of all pyramids.
• Figure 7: $I$ with optimal solution
• Figure 8: $I_\varepsilon$ with optimal solution
• Figure 9: An illustration of FillHoles (Algorithm \ref{['alg:transform']}), post-processing a potential approximate solution for $I_\varepsilon$ to reduce the number of holes.

Theorems & Definitions (16)

• Definition 1: ESfL
• Theorem 2
• Theorem 3
• Corollary 3
• Lemma 4: Li et al. li2020:line
• Theorem 5: Kisfaludi-Bak et al. kisfaludi2021:ptas
• Theorem 5
• Lemma 6: Decomposition
• Lemma 7: Lower Bound
• Definition 8