The Lazy Travelling Salesman Problem in 2

Paz Polak; Gershon Wolansky

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 3, page 538-552
  • ISSN: 1292-8119

Abstract

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We study a parameter (σ) dependent relaxation of the Travelling Salesman Problem on  2 . The relaxed problem is reduced to the Travelling Salesman Problem as σ 0. For increasing σ it is also an ordered clustering algorithm for a set of points in 2 . A dual formulation is introduced, which reduces the problem to a convex optimization, provided the minimizer is in the domain of convexity of the relaxed functional. It is shown that this last condition is generically satisfied, provided σ is large enough.

How to cite

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Polak, Paz, and Wolansky, Gershon. "The Lazy Travelling Salesman Problem in $\mathbb{R}^2$." ESAIM: Control, Optimisation and Calculus of Variations 13.3 (2007): 538-552. <http://eudml.org/doc/249932>.

@article{Polak2007,
abstract = { We study a parameter (σ) dependent relaxation of the Travelling Salesman Problem on $\mathbb\{R\}^2$. The relaxed problem is reduced to the Travelling Salesman Problem as $\sigma\rightarrow$ 0. For increasing σ it is also an ordered clustering algorithm for a set of points in $\mathbb\{R\}^2$. A dual formulation is introduced, which reduces the problem to a convex optimization, provided the minimizer is in the domain of convexity of the relaxed functional. It is shown that this last condition is generically satisfied, provided σ is large enough. },
author = {Polak, Paz, Wolansky, Gershon},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Travelling Salesman Problem; Legendre-Fenchel transform; travelling salesman problem},
language = {eng},
month = {6},
number = {3},
pages = {538-552},
publisher = {EDP Sciences},
title = {The Lazy Travelling Salesman Problem in $\mathbb\{R\}^2$},
url = {http://eudml.org/doc/249932},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Polak, Paz
AU - Wolansky, Gershon
TI - The Lazy Travelling Salesman Problem in $\mathbb{R}^2$
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/6//
PB - EDP Sciences
VL - 13
IS - 3
SP - 538
EP - 552
AB - We study a parameter (σ) dependent relaxation of the Travelling Salesman Problem on $\mathbb{R}^2$. The relaxed problem is reduced to the Travelling Salesman Problem as $\sigma\rightarrow$ 0. For increasing σ it is also an ordered clustering algorithm for a set of points in $\mathbb{R}^2$. A dual formulation is introduced, which reduces the problem to a convex optimization, provided the minimizer is in the domain of convexity of the relaxed functional. It is shown that this last condition is generically satisfied, provided σ is large enough.
LA - eng
KW - Travelling Salesman Problem; Legendre-Fenchel transform; travelling salesman problem
UR - http://eudml.org/doc/249932
ER -

References

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  10. D. Tsafrir, I. Tsafrir, L. Ein-Dor, O. Zuk, D.A. Notterman and E. Domany, Sorting points into neighborhoods (spin): data analysis and visualization by ordering distance matrices. Bioinformatics21 (2005) 2301–2308.  
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