Maximum Traveling Salesman Problem: Difference between revisions

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Created page with "Title: Maximum Traveling Salesman Problem Research Question: How can we efficiently find the longest possible path through a set of points in a geometric space, given different distance norms? Methodology: The researchers considered two classes of norms: polyhedral norms and geometric norms. Polyhedral norms are determined by a centrally-symmetric polyhedron that defines the unit ball, while geometric norms include Euclidean and Lp norms. They proposed algorithms to so..."
 
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Title: Maximum Traveling Salesman Problem
Title: Maximum Traveling Salesman Problem


Research Question: How can we efficiently find the longest possible path through a set of points in a geometric space, given different distance norms?
Research Question: How can we efficiently find the longest possible path through a set of points in a geometric space, considering different distance norms?


Methodology: The researchers considered two classes of norms: polyhedral norms and geometric norms. Polyhedral norms are determined by a centrally-symmetric polyhedron that defines the unit ball, while geometric norms include Euclidean and Lp norms. They proposed algorithms to solve the Maximum Traveling Salesman Problem (MTSP) for both classes.
Methodology: The researchers considered two classes of distance norms: polyhedral norms and geometric norms. They proposed algorithms to solve the Maximum Traveling Salesman Problem (MTSP) for each class.


Results: For polyhedral norms, they showed that the problem can be solved in polynomial time. They presented a simple algorithm with O(n) running time for 2-dimensional metrics, which does not require indirect addressing and remains effective even in comparison-based models. For geometric norms, they proved that the MTSP is NP-hard for Euclidean distances in Rd for d ≥ 3, shedding new light on the difficulties of Euclidean distances.
Results:


Implications: The results suggest that polyhedral norms allow for faster algorithms to solve the MTSP, while the hardness of the problem for geometric norms, particularly Euclidean distances, highlights the complexity of this problem. The simple algorithm for 2-dimensional metrics without indirect addressing provides an intuitive understanding of why polyhedral norms enable fast algorithms.
1. For polyhedral norms, they showed that the problem can be solved in polynomial time. They provided an O(nf−2logn) algorithm, which is particularly efficient for norms with a small number of facets.


Link to Article: https://arxiv.org/abs/0204024v1
2. For the special case of two-dimensional metrics with four facets (which includes the Rectilinear and Sup norms), they presented a simple algorithm with O(n) running time. This algorithm does not require indirect addressing, making it efficient even in comparison-based models.
 
3. They also proved that for the case of Euclidean distances in three-dimensional or higher spaces, the MTSP is NP-hard.
 
Implications: These results provide insights into the complexity of the MTSP for different distance norms. The efficient algorithms for polyhedral norms open up possibilities for solving practical problems in areas such as logistics, routing, and network optimization. The NP-hardness result for Euclidean distances highlights the challenges in solving such problems in higher dimensions.
 
Link to Article: https://arxiv.org/abs/0204024v2
Authors:  
Authors:  
arXiv ID: 0204024v1
arXiv ID: 0204024v2


[[Category:Computer Science]]
[[Category:Computer Science]]
[[Category:Norms]]
[[Category:Norms]]
[[Category:They]]
[[Category:Problem]]
[[Category:Problem]]
[[Category:Geometric]]
[[Category:Distance]]
[[Category:Polyhedral]]
[[Category:Polyhedral]]
[[Category:Euclidean]]

Latest revision as of 04:29, 24 December 2023

Title: Maximum Traveling Salesman Problem

Research Question: How can we efficiently find the longest possible path through a set of points in a geometric space, considering different distance norms?

Methodology: The researchers considered two classes of distance norms: polyhedral norms and geometric norms. They proposed algorithms to solve the Maximum Traveling Salesman Problem (MTSP) for each class.

Results:

1. For polyhedral norms, they showed that the problem can be solved in polynomial time. They provided an O(nf−2logn) algorithm, which is particularly efficient for norms with a small number of facets.

2. For the special case of two-dimensional metrics with four facets (which includes the Rectilinear and Sup norms), they presented a simple algorithm with O(n) running time. This algorithm does not require indirect addressing, making it efficient even in comparison-based models.

3. They also proved that for the case of Euclidean distances in three-dimensional or higher spaces, the MTSP is NP-hard.

Implications: These results provide insights into the complexity of the MTSP for different distance norms. The efficient algorithms for polyhedral norms open up possibilities for solving practical problems in areas such as logistics, routing, and network optimization. The NP-hardness result for Euclidean distances highlights the challenges in solving such problems in higher dimensions.

Link to Article: https://arxiv.org/abs/0204024v2 Authors: arXiv ID: 0204024v2

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