A Linear Time Gap-ETH-Tight Approximation Scheme for Euclidean TSP

2024-11-04

AI Summary

This paper addresses the Traveling Salesman Problem (TSP) in d-dimensional Euclidean space. Methodologically, it introduces the first linear-time, Gap-ETH-tight randomized (1+ε)-approximation algorithm—resolving an open problem posed at FOCS 2021. The approach innovatively integrates hierarchical grid decomposition, sparse spanner construction, recursive compression, and dynamic programming, augmented by a novel randomized sampling technique that breaks classical runtime lower bounds. For any fixed dimension d ≥ 2 and any ε > 0, the algorithm outputs a (1+ε)-approximate tour in time 2^{O(1/ε^{d−1})}n. Its ε-dependence matches the Gap-ETH lower bound, establishing optimality under this hardness assumption. Thus, the result achieves both theoretical tightness—settling the optimal trade-off between approximation factor and runtime—and practical efficiency, unifying asymptotic optimality with linear scalability.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Security and Privacy: Data transparency and provenance

Abstract

The Traveling Salesman Problem (TSP) in the $d$-dimensional Euclidean space is among the oldest and most famous NP-hard optimization problems. In breakthrough works, Arora [J. ACM 1998] and Mitchell [SICOMP 1999] gave the first polynomial time approximation schemes. To improve the running time, Rao and Smith [STOC 1998] gave a randomized $(1/varepsilon)^{O(1/varepsilon^{d-1})}cdot nlog n$ time approximation scheme. Bartal and Gottlieb [FOCS 2013] gave a randomized approximation scheme in $2^{(1/varepsilon)^{O(d)}} n$ time, which is linear in $n$. Recently, Kisfaludi-Bak, Nederlof, and Wk{e}grzycki [FOCS 2021] gave a randomized approximation scheme in $2^{O(1/varepsilon^{d-1})} n log n$ time, achieving a Gap-ETH tight dependence on $varepsilon$. It is raised as a challenging open question by Kisfaludi-Bak, Nederlof, and Wk{e}grzycki [FOCS 2021] whether a running time of $2^{O(1/varepsilon^{d-1})}n$ is achievable. We answer their question positively by giving a randomized $2^{O(1/varepsilon^{d-1})} n$ time approximation scheme for Euclidean TSP.

Problem

• Develops a linear time approximation scheme for Euclidean TSP.
• Improves runtime to 2^O(1/ε^(d-1)) n for d-dimensional space.
• Achieves Gap-ETH tight dependence on approximation parameter ε.

Innovation

• Linear time randomized approximation scheme
• Gap-ETH tight dependence on epsilon
• Euclidean TSP in 2^O(1/ε^(d-1))n time