[논문 리뷰] Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

Matthias Englert, Heiko Röglin외 1명|Warwick Research Archive Portal (University of Warwick)|2023. 02. 14.

Vehicle Routing Optimization Methods참고 문헌 13인용 수 33

한 줄 요약

본 논문은 Euclidean 공간 및 Lp 공간에서 2-Opt의 지수적 하한을 입증하고, φ-perturbed 입력 모델 하에서 2-Opt 개선의 확률적 상한을 도출한다.

ABSTRACT

2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on real world Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every $p\in\mathbb{N}$, a family of $L_p$ instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which $n$ points are placed uniformly at random in the unit square $[0,1]^2$. We consider a more advanced model in which the points can be placed independently according to general distributions on $[0,1]^d$, for an arbitrary $d\ge 2$. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number $n$ of points and the maximal density $ϕ$ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of $ ilde{O}(n^{4+1/3}\cdotϕ^{8/3})$. When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to $ ilde{O}(n^{4+1/3-1/d}\cdotϕ^{8/3})$. If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by $ ilde{O}(n^{4-1/d}\cdotϕ)$. In addition, we prove an upper bound of $O(\sqrt[d]ϕ)$ on the expected approximation factor with respect to all $L_p$ metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with $ϕ=1$ and a smoothed analysis with Gaussian perturbations of standard deviation $σ$ with $ϕ\sim1/σ^d$.

연구 동기 및 목표

• Demonstrate that 2-Opt can take exponentially many steps on Euclidean plane TSP instances.
• Analyze 2-Opt performance under a generalized probabilistic input model with bounded density φ.
• Provide bounds on the expected number of 2-Opt improvements and on the expected approximation ratio under perturbations.
• Extend previous results to φ-perturbed Manhattan and other Lp metrics and relate to smoothed analysis.

제안 방법

• Construct exponential-length 2-Opt improvement sequences using gadget-based instances in the Euclidean plane and general Lp metrics.
• Define and utilize a φ-perturbed input model where each point has density bounded by φ.
• Prove lower bounds by embedding gadget sequences that force exponential state changes.
• Derive upper bounds on the expected length of longest 2-Opt paths under φ-perturbed inputs for Manhattan and Euclidean metrics.
• Show that inserting heuristics start tours can reduce the expected number of steps via Theorem 3.
• Analyze the expected approximation factor of the worst locally optimal tour under φ-perturbed inputs (Theorem 4).

실험 결과

연구 질문

• RQ1 Can 2-Opt exhibit exponential-length improvement sequences in the Euclidean plane?
• RQ2 How does the 2-Opt performance change under a generalized probabilistic input model with density bound φ?
• RQ3 What are the upper bounds on the expected number of 2-Opt improvements for φ-perturbed Manhattan and Euclidean instances?
• RQ4 How does starting from insertion-heuristic tours affect the expected running time and approximation quality of 2-Opt?
• RQ5 What is the expected approximation ratio of the worst 2-Opt local optimum under φ-perturbed inputs across Lp metrics?

주요 결과

• For every p in {1,2,3,...,∞} and n, there exists a 2D Lp TSP instance with 16n vertices whose 2-Opt state graph contains a path of length 2^{n+4}-22.
• The expected length of the longest 2-Opt path is O(n^{4+1/3} φ^{8/3} log(nφ)) for φ-perturbed Euclidean instances with n points.
• The expected length is O(n^{4} φ) for φ-perturbed Manhattan instances with n points.
• With insertion heuristic initialization, the expected number of 2-Opt steps on φ-perturbed Manhattan instances is O(n^{4-1/d} log n · φ).
• The expected approximation ratio of the worst locally optimal 2-Opt tour is O(φ^{1/d}) for φ-perturbed Lp instances.