Maximum Scatter TSP in Doubling Metrics

Kozma, L.; Mömke, Tobias

doi:10.1137/1.9781611974782.10

Cited by 9 publications

(8 citation statements)

References 36 publications

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“…(Such instances of MV-TSP arise e.g. in the Maximum Scatter TSP application by Kozma and Mömke [37]. )…”

Section: Discussionmentioning

confidence: 99%

“…(For instance, we may snap input points to nearby grid points, if doing so does not significantly affect the objective cost.) Recently, this technique was used by Kozma and Mömke, to give an efficient polynomial-time approximation scheme (EPTAS) for the Maximum Scatter TSP in doubling metrics [37], addressing an open question of Arkin et al [4]. In this case, the reduced problem is exactly the MV-TSP.…”

Section: Introductionmentioning

confidence: 99%

“…Our result leads to improvements in applications where MV-TSP is solved as a subroutine. For instance, as a corollary of Theorem 1.1, the approximation scheme for Maximum Scatter TSP [37] can now be implemented in space polynomial in the error parameter ε.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A time- and space-optimal algorithm for the many-visits TSP

Kozma¹,

Mnich²,

Vincze³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of n cities that visits each city c a prescribed number k c of times. Travel costs may be asymmetric, and visiting a city twice in a row may incur a non-zero cost. The MV-TSP problem finds applications in scheduling, geometric approximation, and Hamiltonicity of certain graph families.The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou (SICOMP, 1984). It runs in time n O(n) + O(n 3 log c k c ) and requires n O(n) space. An interesting feature of the Cosmadakis-Papadimitriou algorithm is its logarithmic dependence on the total length c k c of the tour, allowing the algorithm to handle instances with very long tours, beyond what is tractable in the standard TSP setting. However, its superexponential dependence on the number of cities in both its time and space complexity renders the algorithm impractical for all but the narrowest range of this parameter.In this paper we significantly improve on the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time 2 O(n) , i.e. single-exponential in the number of cities, with polynomial space. The space requirement of our algorithm is (essentially) the size of the output, and assuming the Exponential-time Hypothesis (ETH), the time requirement is optimal. Our algorithm is deterministic, and arguably both simpler and easier to analyse than the original approach of Cosmadakis and Papadimitriou. It involves an optimization over directed spanning trees and a recursive, centroid-based decomposition of trees. * A preliminary version of this paper appeared in the ACM-SIAM Symposium on Discrete Algorithms 2019.

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“…(Such instances of MV-TSP arise e.g. in the Maximum Scatter TSP application by Kozma and Mömke [37]. )…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A time- and space-optimal algorithm for the many-visits TSP

Kozma¹,

Mnich²,

Vincze³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Kozma and Mömke [36] showed that there is a PTAS for the MSTSP in doubling spaces of dimension d at most O(log log n), where n is the number of vertices. More precisely, the result is a (1 + ε)-approximation algorithm running in timeÕ 1) .…”

Section: Doubling Spacesmentioning

confidence: 99%

A Modern View on Stability of Approximation

Klasing

Mömke

2018

Lecture Notes in Computer Science

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In order to attack hard optimization problems that do not admit any polynomial-time approximation scheme (PTAS) or α-approximation algorithm for a reasonable constant α (or even with a worse approximability), Hromkovi£ et al. [12,31] introduced the notion of Stability of Approximation. The main idea of the stability concept is to try to split the set of all input instances into (potentially innitely many) classes with respect to the achievable approximation ratio. The crucial point in applying this concept is to nd parameters that well capture the diculty of problem instances of the particular hard problem. The concept of stability of approximation turns out to be ubiquitous in the research of approximation algorithms and is applicable beyond approximation. In the literature, however, the relation to stability of approximation has stayed implicit. The purpose of this survey is to take a broader view and to explicitly analyze algorithmic problems and their algorithms with respect to stability of approximation.I would like to express my belief that, in order to make essential progress in algorithmics, one has to move from measuring the problem hardness in a worst-case manner to classifying the hardness of the instances of the investigated algorithmic problems.

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“…(For instance, we may snap input points to nearby grid points, if doing so does not significantly affect the objective cost.) Recently, this technique was used by Kozma and Mömke, to give an efficient polynomialtime approximation scheme (EPTAS) for the Maximum Scatter TSP in doubling metrics [29], addressing an open question of Arkin et al [4]. In this case, the reduced problem is exactly the MV-TSP.…”

Section: Introductionmentioning

confidence: 99%

Time- and Space-optimal Algorithm for the Many-visits TSP

Berger

Kozma

Mnich

et al. 2020

ACM Trans. Algorithms

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The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of n cities that visits each city c a prescribed number k c of times. Travel costs may be asymmetric, and visiting a city twice in a row may incur a non-zero cost. The MV-TSP problem finds applications in scheduling, geometric approximation, and Hamiltonicity of certain graph families. The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou (SICOMP, 1984). It runs in time n O(n) + O(n 3 log ∑ c k c ) and requires n ᶿ(n) space. An interesting feature of the Cosmadakis-Papadimitriou algorithm is its logarithmic dependence on the total length ∑ c k c of the tour, allowing the algorithm to handle instances with very long tours. The superexponential dependence on the number of cities in both the time and space complexity, however, renders the algorithm impractical for all but the narrowest range of this parameter. In this article, we improve upon the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time 2 O(n) , i.e., single-exponential in the number of cities, using polynomial space. The space requirement of our algorithm is (essentially) the size of the output, and assuming the Exponential-Time Hypothesis (ETH), the problem cannot be solved in time 2 o(n) . Our algorithm is deterministic, and arguably both simpler and easier to analyze than the original approach of Cosmadakis and Papadimitriou. It involves an optimization over directed spanning trees and a recursive, centroid-based decomposition of trees.

show abstract

Maximum Scatter TSP in Doubling Metrics

Cited by 9 publications

References 36 publications

A time- and space-optimal algorithm for the many-visits TSP

A time- and space-optimal algorithm for the many-visits TSP

A Modern View on Stability of Approximation

Time- and Space-optimal Algorithm for the Many-visits TSP

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