# A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

@article{KisfaludiBak2022AGA, title={A Gap-ETH-Tight Approximation Scheme for Euclidean TSP}, author={S{\'a}ndor Kisfaludi-Bak and Jesper Nederlof and Karol Wegrzycki}, journal={2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)}, year={2022}, pages={351-362} }

We revisit the classic task of finding the shortest tour of $n$, points in d-dimensional Euclidean space, for any fixed constant $d\geqslant 2$. We determine the optimal dependence on $\varepsilon$ in the running time of an algorithm that computes a $(1+\varepsilon){-}$ approximate tour, under a plausible assumption, Specifically, we give an algorithm that runs in $2^{\mathcal{O}(1/\varepsilon^{d-1})}n\log n$ time. This improves the previously smallest dependence on $\varepsilon$ in the running…

## One Citation

A Practical Algorithm with Performance Guarantees for the Art~Gallery Problem

- Computer ScienceSoCG
- 2021

A one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program guarantees to find the optimal solution for every vision stable polygon.

## References

SHOWING 1-10 OF 65 REFERENCES

Near-linear time approximation schemes for Steiner tree and forest in low-dimensional spaces

- Computer Science, MathematicsSTOC
- 2021

We give an algorithm that computes a (1+є)-approximate Steiner forest in near-linear time n · 2(1/є)O(ddim2) (loglogn)2, where ddim is the doubling dimension of the metric space. This improves upon…

Truly Optimal Euclidean Spanners

- Computer Science, Mathematics2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
- 2019

It is shown that any (1+ε) -spanner must have lightness Ω(ε^-d), and the upper bound on the lightness of the greedy spanner is improved, implying that the greedy (and other) spanners achieve the optimal size.

Geometric spanner networks

- Computer Science
- 2007

This paper presents a model for designing approximation algorithms with spanners based on the algebraic computation-tree model and some examples show how this model can be applied to well-separated pair decomposition.

A Polynomial-Time Approximation Scheme for Euclidean Steiner Forest

- Mathematics, Computer Science2008 49th Annual IEEE Symposium on Foundations of Computer Science
- 2008

A randomized O(n2 log n)-time approximation scheme for the Steiner forest problem in the Euclidean plane that finds a (1 + epsi)- approximation to the minimum-length forest that connects every pair of terminals.

Light Euclidean Steiner Spanners in the Plane

- Mathematics, Computer ScienceSoCG
- 2021

For every finite set of points in the plane and every $\varepsilon>0$ there exists a Euclidean Steiner-spanner of lightness, and this matches the lower bound for $d=2$.

A Framework for Exponential-Time-Hypothesis-Tight Algorithms and Lower Bounds in Geometric Intersection Graphs

- Mathematics, Computer ScienceSIAM J. Comput.
- 2020

We give an algorithmic and lower bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to intersection graphs ...

A PTAS for subset TSP in minor-free graphs

- MathematicsSODA
- 2020

We give the first PTAS for the subset Traveling Salesperson Problem (TSP) in $H$-minor-free graphs. This resolves a long standing open problem in a long line of work on designing PTASes for TSP in…

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2020

This work surveys developments in the area of parameterization and approximation both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.

Light Euclidean Spanners with Steiner Points

- Computer Science, MathematicsESA
- 2020

The FOCS'19 paper of Le and Solomon, culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy $(1+\epsilon)$-spanner in $\mathbb{R}^d$ is…

ETH-tight algorithms for geometric network problems

- Computer Science
- 2019

The final author version and the galley proof are versions of the publication after peer review that features the final layout of the paper including the volume, issue and page numbers.