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Secret Journey Between Cities: How to Find the Magical Way of the Shortest Path?
In the transportation network between cities, people often worry about finding the shortest path. This is not only a theoretical challenge, but also a practical problem in life. Faced with complex urban layouts, how can we quickly find the shortest path? This article will take you deep into the mysteries of the traveling salesman problem and understand its importance in operations research and computer science.
The hustle and bustle of the city and the interweaving of emotions make the distance between every city full of stories. How can this be measured?
Basic concepts of the traveling salesman problem
The Traveling Salesman Problem (TSP) is widely considered to be an NP-hard problem, meaning that the computational cost of finding the optimal solution increases dramatically as the number of cities increases. Simply put, this problem requires finding a shortest path that visits a specified series of cities, and each city must be visited exactly once and finally returns to the starting point.
Historical BackgroundThe roots of TSP are unclear but can be traced back to the 19th century. A similar question was asked in a travel sales brochure from 1832. It was not until the 20th century that mathematicians William Rowan Hamilton and Thomas Kirkman formalized the problem mathematically. In the 1930s, mathematician Carl Menger studied this problem in Vienna and at Harvard University and proposed an explicit brute-force algorithm to solve it.
Computational Complexity and Algorithms
The computational complexity of the traveling salesman problem makes it an important area of study in theoretical computer science. Although this problem is computationally extremely difficult in the best case, many heuristic and exact algorithms have been proposed, some of which can solve the problem for tens of thousands of cities. These algorithms make use of various mathematical tools, including integer linear programming and the method of tangent planes.
“Even in the most complex scenarios, finding the approximate optimal solution is still an important step in the journey.”
Real-world applications
The traveling salesman problem has wide applications in various fields, such as logistics management, microchip manufacturing, and DNA sequencing. In these applications, cities often represent customers or specific locations, while distance can be considered as cost or time. In astronomy, when astronomers are observing multiple celestial bodies, they want to move the telescope in the best way possible, minimizing the movement time.
Similar problems and variants
TSP has many related problems and variants, including symmetric and asymmetric traveling salesman problems, the latter of which considers different distances in different directions. In addition, the bottleneck traveling salesman problem requires finding a Hamiltonian circuit with minimum and maximum edge weights, which is quite important in practical applications. For example, avoid the hassles that large buses can cause on narrow streets.
Minimum spanning trees and approximate algorithms
Creating a minimum spanning tree is an effective technique in finding solutions to the TSP. According to the Christofides algorithm, in the worst case, the length of the obtained solution will not exceed 1.5 times the optimal solution. Although this technology has gained attention over the past few decades, only recently have progressively improved algorithms been developed.
ConclusionThe traveling salesman problem is not only a mathematical challenge, it can also be found everywhere in real life, affecting our daily decisions and actions. As technology develops, will we be able to find more efficient solutions in the future to shorten travel distances between cities?
