Extrema of a Class of Functions on a Finite Set

1974 ◽  
Vol 26 (4) ◽  
pp. 806-819
Author(s):  
Kenneth W. Lebensold
Keyword(s):  
Positive Integer ◽  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Polygonal Path ◽  
Finite Set ◽  
Vertex Set ◽  
The Traveling Salesman Problem ◽  
The Cost ◽  
Special Case ◽  
Class Of Functions

In this paper, we are concerned with the following problem: Let S be a finite set and Sm* ⊂ 2S a collection of subsets of S each of whose members has m elements (m a positive integer). Let f be a real-valued function on S and, for p ∊ Sm*, define f(P) as Σs∊pf (s). We seek the minimum (or maximum) of the function f on the set Sm*.The Traveling Salesman Problem is to find the cheapest polygonal path through a given set of vertices, given the cost of getting from any vertex to any other. It is easily seen that the Traveling Salesman Problem is a special case of this system, where S is the set of all edges joining pairs of points in the vertex set, Sm* is the set of polygons, each polygon has m elements (m = no. of points in the vertex set = no. of edges per polygon), and f is the cost function.

GENERALISATIONS OF THE GILMORE-GOMORY TRAVELING SALESMAN PROBLEM AND THE GILMORE-GOMORY SCHEME: A SURVEY

2001 ◽  
Vol 03 (02n03) ◽  
pp. 213-235 ◽  
Author(s):  
SANTOSH N. KABADI
Keyword(s):  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Major Portion ◽  
Polynomially Solvable ◽  
The Traveling Salesman Problem ◽  
Special Case

One of the first and perhaps the most well-known polynomially solvable special case of the traveling salesman problem (TSP) is the Gilmore-Gomory case (G-G TSP). Gilmore and Gomory presented an interesting patching algorithm for this case with a fairly non-trivial proof of its validity. Their work has motivated a great deal of research in the area leading to various generalisations of their results and thereby identification of fairly large polynomially solvable subclasses of the TSP. These results form a major portion of the literature on solvable cases of the TSP. In this paper, we survey the main results on solvable cases of the TSP which are direct generalisations of the G-G TSP and/or the Gilmore-Gomory patching scheme.

Metaheuristics Approaches for the Travelling Salesman Problem on a Spherical Surface

2021 ◽  
pp. 94-113
Author(s):  
Yusuf Sahin ◽  
Erdal Aydemir ◽  
Kenan Karagul ◽  
Sezai Tokat ◽  
Burhan Oran
Keyword(s):  
Genetic Algorithms ◽  
Traveling Salesman Problem ◽  
Spherical Surface ◽  
Travelling Salesman Problem ◽  
Traveling Salesman ◽  
Heuristic Methods ◽  
Approximate Algorithms ◽  
Travelling Salesman ◽  
The Traveling Salesman Problem ◽  
Special Case

Traveling salesman problem in which all the vertices are assumed to be on a spherical surface is a special case of the conventional travelling salesman problem. There are exact and approximate algorithms for the travelling salesman problem. As the solution time is a performance parameter in most real-time applications, approximate algorithms always have an important area of research for both researchers and engineers. In this chapter, approximate algorithms based on heuristic methods are considered for the travelling salesman problem on the sphere. Firstly, 28 test instances were newly generated on the unit sphere. Then, using various heuristic methods such as genetic algorithms, ant colony optimization, and fluid genetic algorithms, the initial solutions for solving test instances of the traveling salesman problem are obtained in Matlab®. Then, the initial heuristic solutions are used as input for the 2-opt algorithm. The performances and time complexities of the applied methods are analyzed as a conclusion.

scholarly journals A Revisiting Method Using a Covariance Traveling Salesman Problem Algorithm for Landmark-Based Simultaneous Localization and Mapping

Sensors ◽  
2019 ◽  
Vol 19 (22) ◽  
pp. 4910 ◽  
Author(s):  
Hyejeong Ryu
Keyword(s):  
Measurement Uncertainty ◽  
Traveling Salesman Problem ◽  
Optimal Path ◽  
Simultaneous Localization And Mapping ◽  
Traveling Salesman ◽  
Localization And Mapping ◽  
Sensor Measurement ◽  
Slam Algorithm ◽  
The Traveling Salesman Problem ◽  
The Cost

This paper presents an efficient revisiting algorithm for landmark-based simultaneous localization and mapping (SLAM). To reduce SLAM uncertainty in terms of a robot’s pose and landmark positions, the method autonomously evaluates valuable landmarks for the data associations in the SLAM algorithm and selects positions to revisit by considering both landmark visibility and sensor measurement uncertainty. The optimal path among the selected positions is obtained by applying the traveling salesman problem (TSP) algorithm. To plan a path that reduces overall uncertainty, the cost matrix associated with the change in covariance between all selected positions of all pairs is applied for the TSP algorithm. From simulations, it is verified that the proposed method efficiently reduces and maintains SLAM uncertainty at the low level compared to the backtracking method.

scholarly journals A Graph-Based Approach to Optimal Scan Chain Stitching Using RTL Design Descriptions

VLSI Design ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Lilia Zaourar ◽  
Yann Kieffer ◽  
Chouki Aktouf
Keyword(s):  
Approximation Algorithms ◽  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Graph Models ◽  
Rtl Design ◽  
And Performance ◽  
Scan Chain ◽  
The Traveling Salesman Problem ◽  
The Cost

The scan chain insertion problem is one of the mandatory logic insertion design tasks. The scanning of designs is a very efficient way of improving their testability. But it does impact size and performance, depending on the stitching ordering of the scan chain. In this paper, we propose a graph-based approach to a stitching algorithm for automatic and optimal scan chain insertion at the RTL. Our method is divided into two main steps. The first one builds graph models for inferring logical proximity information from the design, and then the second one uses classic approximation algorithms for the traveling salesman problem to determine the best scan-stitching ordering. We show how this algorithm allows the decrease of the cost of both scan analysis and implementation, by measuring total wirelength on placed and routed benchmark designs, both academic and industrial.

scholarly journals A generalization of the convex-hull-and-line traveling salesman problem

1998 ◽  
Vol 2 (2) ◽  
pp. 177-191 ◽  
Author(s):  
Md. Fazle Baki ◽  
S. N. Kabadi
Keyword(s):  
Convex Hull ◽  
Traveling Salesman Problem ◽  
Equivalence Relation ◽  
General Class ◽  
Traveling Salesman ◽  
Cost Matrix ◽  
Solvable Case ◽  
Special Cases ◽  
The Traveling Salesman Problem ◽  
The Cost

Two instances of the traveling salesman problem, on the same node set {1,2,…,n} but with different cost matrices C and C′ , are equivalent iff there exist {ai,bi: i=1,…, n} such that for any 1≤i, j≤n,i≠j,C′(i,j)=C(i,j)+ai+bj [7]. One of the well-solved special cases of the traveling salesman problem (TSP) is the convex-hull-and-line TSP. We extend the solution scheme for this class of TSP given in [9] to a more general class which is closed with respect to the above equivalence relation. The cost matrix in our general class is a certain composition of Kalmanson matrices. This gives a new, non-trivial solvable case of TSP.

A New Heuristic Procedure To Solve The Traveling Salesman Problem

2007 ◽  
Vol 5 (1) ◽  
pp. 1-9
Author(s):  
Paulo Henrique Siqueira ◽  
Sérgio Scheer ◽  
Maria Teresinha Arns Steiner
Keyword(s):  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Heuristic Procedure ◽  
The Traveling Salesman Problem

scholarly journals An Improved Whale Optimization Algorithm for the Traveling Salesman Problem

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 48
Author(s):  
Jin Zhang ◽  
Li Hong ◽  
Qing Liu
Keyword(s):  
Traveling Salesman Problem ◽  
Optimization Algorithm ◽  
Optimization Problems ◽  
Relevant Literature ◽  
Population Diversity ◽  
Traveling Salesman ◽  
Whale Optimization Algorithm ◽  
Neighborhood Search ◽  
Whale Optimization ◽  
The Traveling Salesman Problem

The whale optimization algorithm is a new type of swarm intelligence bionic optimization algorithm, which has achieved good optimization results in solving continuous optimization problems. However, it has less application in discrete optimization problems. A variable neighborhood discrete whale optimization algorithm for the traveling salesman problem (TSP) is studied in this paper. The discrete code is designed first, and then the adaptive weight, Gaussian disturbance, and variable neighborhood search strategy are introduced, so that the population diversity and the global search ability of the algorithm are improved. The proposed algorithm is tested by 12 classic problems of the Traveling Salesman Problem Library (TSPLIB). Experiment results show that the proposed algorithm has better optimization performance and higher efficiency compared with other popular algorithms and relevant literature.

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