scholarly journals How to Solve the Traveling Salesman Problem

2021 ◽  
Author(s):  
Weiqi Li
Keyword(s):  
Traveling Salesman Problem ◽  
Solution Space ◽  
Traveling Salesman ◽  
Optimal Solutions ◽  
Computational Power ◽  
Critical Question ◽  
Search System ◽  
Combinatorial Explosion ◽  
Intractable Problem ◽  
The Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is believed to be an intractable problem and have no practically efficient algorithm to solve it. The intrinsic difficulty of the TSP is associated with the combinatorial explosion of potential solutions in the solution space. When a TSP instance is large, the number of possible solutions in the solution space is so large as to forbid an exhaustive search for the optimal solutions. The seemingly “limitless” increase of computational power will not resolve its genuine intractability. Do we need to explore all the possibilities in the solution space to find the optimal solutions? This chapter offers a novel perspective trying to overcome the combinatorial complexity of the TSP. When we design an algorithm to solve an optimization problem, we usually ask the critical question: “How can we find all exact optimal solutions and how do we know that they are optimal in the solution space?” This chapter introduces the Attractor-Based Search System (ABSS) that is specifically designed for the TSP. This chapter explains how the ABSS answer this critical question. The computing complexity of the ABSS is also discussed.

Solution Attractor of Local Search System: A Method to Reduce Computational Complexity of the Traveling Salesman Problem

2020 ◽  
Author(s):  
Weiqi Li
Keyword(s):  
Computational Complexity ◽  
Local Search ◽  
Traveling Salesman Problem ◽  
Search Algorithm ◽  
Optimal Solution ◽  
Solution Space ◽  
Search Space ◽  
Traveling Salesman ◽  
Search System ◽  
The Traveling Salesman Problem

The traveling salesman problem (TSP) is presumably difficult to solve exactly using local search algorithms. It can be exactly solved by only one algorithm—the enumerative search algorithm. However, the scanning of all possible solutions requires exponential computing time. Do we need exploring all the possibilities to find the optimal solution? How can we narrow down the search space effectively and efficiently for an exhausted search? This chapter attempts to answer these questions. A local search algorithm is a discrete dynamical system, in which a search trajectory searches a part of the solution space and stops at a locally optimal point. A solution attractor of a local search system for the TSP is defined as a subset of the solution space that contains all locally optimal tours. The solution attractor concept gives us great insight into the computational complexity of the TSP. If we know where the solution attractor is located in the solution space, we simply completely search the solution attractor, rather than the entire solution space, to find the globally optimal tour. This chapter describes the solution attractor of local search system for the TSP and then presents a novel search system—the attractor-based search system—that can solve the TSP much efficiently with global optimality guarantee.

scholarly journals On the Traveling Salesman Problem in Nautical Environments: an Evolutionary Computing Approach to Optimization of Tourist Route Paths in Medulin, Croatia

2019 ◽  
Vol 57 (1) ◽  
pp. 71-87 ◽  
Author(s):  
Sandi Baressi Šegota ◽  
Ivan Lorencin ◽  
Kazuhiro Ohkura ◽  
Zlatan Car
Keyword(s):  
Traveling Salesman Problem ◽  
Optimal Path ◽  
Solution Space ◽  
Evolutionary Computing ◽  
Problem Formulation ◽  
Computing Methods ◽  
Traveling Salesman ◽  
Planning Problem ◽  
Extensive Search ◽  
The Traveling Salesman Problem

The Traveling salesman problem (TSP) defines the problem of finding the optimal path between multiple points, connected by paths of a certain cost. This paper applies that problem formulation in the maritime environment, specifically a path planning problem for a tour boat visiting popular tourist locations in Medulin, Croatia. The problem is solved using two evolutionary computing methods – the genetic algorithm (GA) and the simulated annealing (SA) - and comparing the results (are compared) by an extensive search of the solution space. The results show that evolutionary computing algorithms provide comparable results to an extensive search in a shorter amount of time, with SA providing better results of the two.

scholarly journals Advanced Harmony Search with Ant Colony Optimization for Solving the Traveling Salesman Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Ho-Yoeng Yun ◽  
Suk-Jae Jeong ◽  
Kyung-Sup Kim
Keyword(s):  
Ant Colony Optimization ◽  
Traveling Salesman Problem ◽  
Harmony Search ◽  
Solution Space ◽  
Global Optimum ◽  
Traveling Salesman ◽  
Ant Colony ◽  
Local Optimum ◽  
Optimum Solution ◽  
The Traveling Salesman Problem

We propose a novel heuristic algorithm based on the methods of advanced Harmony Search and Ant Colony Optimization (AHS-ACO) to effectively solve the Traveling Salesman Problem (TSP). The TSP, in general, is well known as an NP-complete problem, whose computational complexity increases exponentially by increasing the number of cities. In our algorithm, Ant Colony Optimization (ACO) is used to search the local optimum in the solution space, followed by the use of the Harmony Search to escape the local optimum determined by the ACO and to move towards a global optimum. Experiments were performed to validate the efficiency of our algorithm through a comparison with other algorithms and the optimum solutions presented in the TSPLIB. The results indicate that our algorithm is capable of generating the optimum solution for most instances in the TSPLIB; moreover, our algorithm found better solutions in two cases (kroB100 and pr144) when compared with the optimum solution presented in the TSPLIB.

New Genetic Operator (Jump Crossover) for the Traveling Salesman Problem

2016 ◽  
pp. 1739-1752 ◽  
Author(s):  
Hicham El Hassani ◽  
Said Benkachcha ◽  
Jamal Benhra
Keyword(s):  
Traveling Salesman Problem ◽  
Optimization Methods ◽  
Search Space ◽  
Traveling Salesman ◽  
Optimal Solutions ◽  
Solution Quality ◽  
Hard Problems ◽  
The Traveling Salesman Problem ◽  
Np Hard Problems

Inspired by nature, genetic algorithms (GA) are among the greatest meta-heuristics optimization methods that have proved their effectiveness to conventional NP-hard problems, especially the traveling salesman problem (TSP) which is one of the most studied supply chain management problems. This paper proposes a new crossover operator called Jump Crossover (JMPX) for solving the travelling salesmen problem using a genetic algorithm (GA) for near-optimal solutions, to conclude on its efficiency compared to solutions quality given by other conventional operators to the same problem, namely, Partially matched crossover (PMX), Edge recombination Crossover (ERX) and r-opt heuristic with consideration of computational overload. The authors adopt a low mutation rate to isolate the search space exploration ability of each crossover. The experimental results show that in most cases JMPX can remarkably improve the solution quality of the GA compared to the two existing classic crossover approaches and the r-opt heuristic.

New Genetic Operator (Jump Crossover) for the Traveling Salesman Problem

2015 ◽  
Vol 6 (2) ◽  
pp. 33-44 ◽  
Author(s):  
Hicham El Hassani ◽  
Said Benkachcha ◽  
Jamal Benhra
Keyword(s):  
Traveling Salesman Problem ◽  
Optimization Methods ◽  
Search Space ◽  
Traveling Salesman ◽  
Optimal Solutions ◽  
Crossover Operator ◽  
Solution Quality ◽  
Hard Problems ◽  
The Traveling Salesman Problem

Inspired by nature, genetic algorithms (GA) are among the greatest meta-heuristics optimization methods that have proved their effectiveness to conventional NP-hard problems, especially the traveling salesman problem (TSP) which is one of the most studied supply chain management problems. This paper proposes a new crossover operator called Jump Crossover (JMPX) for solving the travelling salesmen problem using a genetic algorithm (GA) for near-optimal solutions, to conclude on its efficiency compared to solutions quality given by other conventional operators to the same problem, namely, Partially matched crossover (PMX), Edge recombination Crossover (ERX) and r-opt heuristic with consideration of computational overload. The authors adopt a low mutation rate to isolate the search space exploration ability of each crossover. The experimental results show that in most cases JMPX can remarkably improve the solution quality of the GA compared to the two existing classic crossover approaches and the r-opt heuristic.

scholarly journals Algorithmic-Information Theory interpretation to the Traveling Salesman Problem

2019 ◽  
Author(s):  
Matheus Santana Lima
Keyword(s):  
Information Theory ◽  
Traveling Salesman Problem ◽  
Time Complexity ◽  
Communication Systems ◽  
Communication Channel ◽  
Transmission Rate ◽  
Solution Space ◽  
Traveling Salesman ◽  
Computational Time ◽  
The Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is a important optimization problem in computer science, mathematics and logistics. It belongs to the class of NP-Hard problems and can be very time consuming to find solutions to large instances with guarantee optimality. As number of city-nodes in the graph increases, the amount of valid route tours also growths rapidly and thus requiring considerable time to evaluate and classify each permutation. The objective of the heuristic process is to search the solution space for the optimal solution while maximizing the attached utility-cost function (i.e. finding the shortest euclidean distance tour) and minimizing the computational time complexity of the algorithm.Many complex real world scenarios can be reduced to a simulation of a salesman trying to find the shortest (length) Hamiltonian (cycle) route in a euclidean super-graph G*. If each city-node is modeled as a input symbol in a communication channel represented by an output pair with consistent probabilities distribution thus an polynomial-time probabilistic algorithm can use this information to improve the solution quality at the same rate of transmission of information over the channel.In this paper we explore an quantitative stochastic process based in Algorithm Information Theory and the Shannon-Kelly criterion to find valid near optimal solutions using a new growth- optimal strategy applied to the TSP problem that have statistically significant transmission rate even when no encoding scheme is available, regardless of time-complexity of the problem.Previous heuristics such as 2 opt, Genetic Algorithms (GA) and Simulated Annealing (SA) approach’s the TSP problem by relying on a priori knowledge about the data distribution in order to reduce the probability of error in finding the best candidate solution tour.In this work we propose a method that models the solution space boundaries of the TSP problem as a communication channel by means of Information Theory. We describe a search algorithm that check for patterns (i.e information content) in the elements of a constrained solution space modeled as messages transmitted through communication systems. The boundaries of the search space are defined by the Kolmogorov complexity of the candidate solutions sequences. We conclude with an discussion about the quality of the results and implications for general decision problem in Turing machines.

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
Sign in / Sign up

Export Citation Format

Share Document