Reducing the Size of Traveling Salesman Problem Instances by Fixing Edges

2007 ◽  
pp. 72-83 ◽  
Author(s):  
Thomas Fischer ◽  
Peter Merz
Keyword(s):  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Problem Instances

scholarly journals Simple Constructive, Insertion, and Improvement Heuristics Based on the Girding Polygon for the Euclidean Traveling Salesman Problem

Algorithms ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 5 ◽  
Author(s):  
Víctor Pacheco-Valencia ◽  
José Alberto Hernández ◽  
José María Sigarreta ◽  
Nodari Vakhania
Keyword(s):  
Traveling Salesman Problem ◽  
Real Life ◽  
Traveling Salesman ◽  
Computational Time ◽  
Problem Instance ◽  
Dimensional Euclidean Space ◽  
Benchmark Problem ◽  
Small Constant ◽  
Euclidean Traveling Salesman Problem ◽  
Problem Instances

The Traveling Salesman Problem (TSP) aims at finding the shortest trip for a salesman, who has to visit each of the locations from a given set exactly once, starting and ending at the same location. Here, we consider the Euclidean version of the problem, in which the locations are points in the two-dimensional Euclidean space and the distances are correspondingly Euclidean distances. We propose simple, fast, and easily implementable heuristics that work well, in practice, for large real-life problem instances. The algorithm works on three phases, the constructive, the insertion, and the improvement phases. The first two phases run in time O ( n 2 ) and the number of repetitions in the improvement phase, in practice, is bounded by a small constant. We have tested the practical behavior of our heuristics on the available benchmark problem instances. The approximation provided by our algorithm for the tested benchmark problem instances did not beat best known results. At the same time, comparing the CPU time used by our algorithm with that of the earlier known ones, in about 92% of the cases our algorithm has required less computational time. Our algorithm is also memory efficient: for the largest tested problem instance with 744,710 cities, it has used about 50 MiB, whereas the average memory usage for the remained 217 instances was 1.6 MiB.

scholarly journals Ant Colony Optimization with Warm-Up

Algorithms ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 295
Author(s):  
Mattia Neroni
Keyword(s):  
Ant Colony Optimization ◽  
Traveling Salesman Problem ◽  
Industrial Application ◽  
Optimization Problems ◽  
Traveling Salesman ◽  
Ant Colony ◽  
Warm Up ◽  
Probabilistic Technique ◽  
Starting Point ◽  
Problem Instances

The Ant Colony Optimization (ACO) is a probabilistic technique inspired by the behavior of ants for solving computational problems that may be reduced to finding the best path through a graph. Some species of ants deposit pheromone on the ground to mark some favorable paths that should be used by other members of the colony. Ant colony optimization implements a similar mechanism for solving optimization problems. In this paper a warm-up procedure for the ACO is proposed. During the warm-up, the pheromone matrix is initialized to provide an efficient new starting point for the algorithm, so that it can obtain the same (or better) results with fewer iterations. The warm-up is based exclusively on the graph, which, in most applications, is given and does not need to be recalculated every time before executing the algorithm. In this way, it can be made only once, and it speeds up the algorithm every time it is used from then on. The proposed solution is validated on a set of traveling salesman problem instances, and in the simulation of a real industrial application for the routing of pickers in a manual warehouse. During the validation, it is compared with other ACO adopting a pheromone initialization technique, and the results show that, in most cases, the adoption of the proposed warm-up allows the ACO to obtain the same or better results with fewer iterations.

scholarly journals Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem

2004 ◽  
Vol 21 ◽  
pp. 471-497 ◽  
Author(s):  
W. Zhang
Keyword(s):  
Phase Transitions ◽  
Combinatorial Optimization ◽  
Traveling Salesman Problem ◽  
Assignment Problem ◽  
Combinatorial Problems ◽  
Traveling Salesman ◽  
Study Phase ◽  
Combinatorial Optimization Problems ◽  
Asymmetric Traveling Salesman Problem ◽  
Problem Instances

In recent years, there has been much interest in phase transitions of combinatorial problems. Phase transitions have been successfully used to analyze combinatorial optimization problems, characterize their typical-case features and locate the hardest problem instances. In this paper, we study phase transitions of the asymmetric Traveling Salesman Problem (ATSP), an NP-hard combinatorial optimization problem that has many real-world applications. Using random instances of up to 1,500 cities in which intercity distances are uniformly distributed, we empirically show that many properties of the problem, including the optimal tour cost and backbone size, experience sharp transitions as the precision of intercity distances increases across a critical value. Our experimental results on the costs of the ATSP tours and assignment problem agree with the theoretical result that the asymptotic cost of assignment problem is pi ^2 /6 the number of cities goes to infinity. In addition, we show that the average computational cost of the well-known branch-and-bound subtour elimination algorithm for the problem also exhibits a thrashing behavior, transitioning from easy to difficult as the distance precision increases. These results answer positively an open question regarding the existence of phase transitions in the ATSP, and provide guidance on how difficult ATSP problem instances should be generated.

scholarly journals On Tightly Bounding the Dubins Traveling Salesman's Optimum

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Satyanarayana G. Manyam ◽  
Sivakumar Rathinam
Keyword(s):  
Lower Bound ◽  
Traveling Salesman Problem ◽  
Optimal Solution ◽  
Traveling Salesman ◽  
Motion Constraints ◽  
Euclidean Traveling Salesman Problem ◽  
Problem Instances ◽  
Feasible Solutions ◽  
The Cost ◽  
Dubins Traveling Salesman Problem

The Dubins traveling salesman problem (DTSP) has generated significant interest over the last decade due to its occurrence in several civil and military surveillance applications. This problem requires finding a curvature constrained shortest path for a vehicle visiting a set of target locations. Currently, there is no algorithm that can find an optimal solution to the DTSP. In addition, relaxing the motion constraints and solving the resulting Euclidean traveling salesman problem (ETSP) provide the only lower bound available for the DTSP. However, in many problem instances, the lower bound computed by solving the ETSP is far below the cost of the feasible solutions obtained by some well-known algorithms for the DTSP. This paper addresses this fundamental issue and presents the first systematic procedure for developing tight lower bounds for the DTSP.

scholarly journals Comparing greedy constructive heuristic subtour elimination methods for the traveling salesman problem

2020 ◽  
Vol 4 (2) ◽  
pp. 167-182
Author(s):  
Petar Jackovich ◽  
Bruce Cox ◽  
Raymond R. Hill
Keyword(s):  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Computational Results ◽  
Content Type ◽  
Constructive Heuristic ◽  
Subtour Elimination ◽  
Constructive Heuristics ◽  
Problem Instances ◽  
Feasible Solutions ◽  
The Traveling Salesman Problem

Purpose This paper aims to define the class of fragment constructive heuristics used to compute feasible solutions for the traveling salesman problem (TSP) into edge-greedy and vertex-greedy subclasses. As these subclasses of heuristics can create subtours, two known methodologies for subtour elimination on symmetric instances are reviewed and are expanded to cover asymmetric problem instances. This paper introduces a third novel subtour elimination methodology, the greedy tracker (GT), and compares it to both known methodologies. Design/methodology/approach Computational results for all three subtour elimination methodologies are generated across 17 symmetric instances ranging in size from 29 vertices to 5,934 vertices, as well as 9 asymmetric instances ranging in size from 17 to 443 vertices. Findings The results demonstrate the GT is the fastest method for preventing subtours for instances below 400 vertices. Additionally, a distinction between fragment constructive heuristics and the subtour elimination methodology used to ensure the feasibility of resulting solutions enables the introduction of a new vertex-greedy fragment heuristic called ordered greedy. Originality/value This research has two main contributions: first, it introduces a novel subtour elimination methodology. Second, the research introduces the concept of ordered lists which remaps the TSP into a new space with promising initial computational results.

scholarly journals CUDA Accelerated 2-OPT Local Search for the Traveling Salesman Problem

2020 ◽  
Author(s):  
Donald Davendra ◽  
Magdalena Metlicka ◽  
Magdalena Bialic-Davendra
Keyword(s):  
Local Search ◽  
Traveling Salesman Problem ◽  
High Performance ◽  
Search Algorithm ◽  
Traveling Salesman ◽  
Processing Unit ◽  
Large Problem ◽  
Intractable Problems ◽  
Problem Instances ◽  
The Traveling Salesman Problem

This research involves the development of a compute unified device architecture (CUDA) accelerated 2-opt local search algorithm for the traveling salesman problem (TSP). As one of the fundamental mathematical approaches to solving the TSP problem, the time complexity has generally reduced its efficiency, especially for large problem instances. Graphic processing unit (GPU) programming, especially CUDA has become more mainstream in high-performance computing (HPC) approaches and has made many intractable problems at least reasonably solvable in acceptable time. This chapter describes two CUDA accelerated 2-opt algorithms developed to solve the asymmetric TSP problem. Three separate hardware configurations were used to test the developed algorithms, and the results validate that the execution time decreased significantly, especially for the large problem instances when deployed on the GPU.

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