The malleability of TSP 2Opt

Fischer, Sophie; Torenvliet, Leen

doi:10.1007/3-540-60618-1_73

Cited by 3 publications

(6 citation statements)

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“…Starting with an arbitrary tour, the expected number of steps performed by 2-Opt on φ-perturbed Manhattan instances with n vertices is O(n 6 • log n • φ) if the coordinates of every vertex are drawn independently.…”

Section: Theoremmentioning

confidence: 99%

“…In a φ-perturbed Manhattan instance with n vertices, the probability that there exists a pair of type 0 or type 1 in which both 2-changes are improvements by at most ε is O(n 6…”

Section: Lemma 10mentioning

confidence: 99%

“…As an example let us consider the case d = 2, let the first order σ 1 be x 1 1 5 1 ≤ x 6 1 , and let the second order σ 2 be…”

Section: Lemma 10mentioning

confidence: 99%

“…i , it must be true that x 3 i ≤ x 6 i in order for x 6 i to cancel out, and it must be true that x 5 i ≤ x 1 i in order for x 5 i to cancel out. Hence, either…”

Section: Lemma 11mentioning

confidence: 99%

“…The only result in this regard is due to Krentel [9] who shows that it is PLS-complete to compute a local optimum for the metric TSP for k-Opt for some constant k. It is not known whether his construction can be embedded into the Euclidean metric and whether it is PLS-complete to compute locally optimal solutions for 2-Opt. Fischer and Torenvliet [6] show, however, that for the general TSP, it is PSPACE-hard to compute a local optimum for 2-Opt that is reachable from a given initial tour. The obvious open question concerning the probabilistic analysis is how the gap between experiments and theory can be narrowed further.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

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Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

2013

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Section: Theoremmentioning

confidence: 99%

“…In a φ-perturbed Manhattan instance with n vertices, the probability that there exists a pair of type 0 or type 1 in which both 2-changes are improvements by at most ε is O(n 6…”

Section: Lemma 10mentioning

confidence: 99%

“…As an example let us consider the case d = 2, let the first order σ 1 be x 1 1 5 1 ≤ x 6 1 , and let the second order σ 2 be…”

Section: Lemma 10mentioning

confidence: 99%

“…i , it must be true that x 3 i ≤ x 6 i in order for x 6 i to cancel out, and it must be true that x 5 i ≤ x 1 i in order for x 5 i to cancel out. Hence, either…”

Section: Lemma 11mentioning

confidence: 99%

Section: Conclusion and Open Problemsmentioning

confidence: 99%

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Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

2013

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The Complexity of  CaRet + Chop

Bozzelli

2008

2008 15th International Symposium on Temporal Representation and Reasoning

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We investigate the complexity of satisfiability and pushdown model-checking of the extension of the logic CaRet with the binary regular modality 'Chop'. We present automata-theoretic decision procedures based on a direct and compositional construction, which for finite (resp., infinite) words require time of exponential height equal to the nesting depth of chop modality plus one (resp., plus two). Moreover, we provide lower bounds which match the upper bounds for the case of finite words.

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Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

Englert¹,

Röglin²,

Vöcking³

2023

Preprint

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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 ∈ 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 , where it was shown that the expected number of steps is bounded by Õ(n 10 ) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed independently according to general distributions on [0, 1] d , for an arbitrary d ≥ 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 Õ(n 4+1/3 • φ 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 Õ(n 4+1/3−1/d •φ 8/3 ). If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by Õ(n 4−1/d • φ). In addition, we prove an upper bound of O( 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 ∼ 1/σ d .

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The malleability of TSP 2Opt

Cited by 3 publications

References 6 publications

Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

The Complexity of  CaRet + Chop

Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

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The malleability of TSP 2Opt

Cited by 3 publications

References 6 publications

Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

The Complexity of&#x0A0;&#x0A0;CaRet + Chop

Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

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The Complexity of CaRet + Chop