Branch and Bound Methods for the Traveling Salesman Problem

1983 ◽  
Author(s):  
Egon Balas ◽  
Paolo Toth
Keyword(s):  
Traveling Salesman Problem ◽  
Branch And Bound ◽  
Traveling Salesman ◽  
Branch And Bound Methods ◽  
The Traveling Salesman Problem

Implementing Parallel Branch-and-Bound with Extended Sisal 2.0

1998 ◽  
Vol 08 (01) ◽  
pp. 41-50
Author(s):  
Yung-Syau Chen ◽  
Jean-Luc Gaudiot
Keyword(s):  
Traveling Salesman Problem ◽  
Branch And Bound ◽  
Optimization Technique ◽  
Traveling Salesman ◽  
Functional Language ◽  
Global Variables ◽  
Parallel Version ◽  
Parallel Branch And Bound ◽  
Hard Problems ◽  
The Traveling Salesman Problem

Parallel branch-and-bound is an optimization technique which renders more efficient the solution of some hard problems such as the puzzle of colored blocks and the traveling-salesman problem. In a functional language such as Sisal 2.0, it is difficult for the programmer to describe a parallel version of this technique due to the lack of imperative features in the language. In this paper, we propose a version of Sisal 2.0 extended with user-declared mutable global variables in order to enable Sisal programmers to apply the parallel branch-and-bound technique. In a simple example (the puzzle of colored blocks), we show that this approach yields better performance than either conventional functional programs or imperative programs. It is easy to see that the same strategy can be used to solve a number of hard problems such as the traveling-salesman problem.

scholarly journals On the Classical Version of the Branch and Bound Method

2021 ◽  
pp. 21-44
Author(s):  
Boris Melnikov ◽  
◽  
Elena Melnikova ◽  
Keyword(s):  
Traveling Salesman Problem ◽  
Branch And Bound ◽  
Discrete Optimization ◽  
Optimization Problems ◽  
Traveling Salesman ◽  
Branch And Bound Method ◽  
Main Subject ◽  
Classical Version ◽  
The Traveling Salesman Problem ◽  
Discrete Optimization Problems

In the computer literature, a lot of problems are described that can be called discrete optimization problems: from encrypting information on the Internet (including creating programs for digital cryptocurrencies) before searching for “interests” groups in social networks. Often, these problems are very difficult to solve on a computer, hence they are called “intractable”. More precisely, the possible approaches to quickly solving these problems are difficult to solve (to describe algorithms, to program); the brute force solution, as a rule, is programmed simply, but the corresponding program works much slower. Almost every one of these intractable problems can be called a mathematical model. At the same time, both the model itself and the algorithms designed to solve it are often created for one subject area, but they can also be used in many other areas. An example of such a model is the traveling salesman problem. The peculiarity of the problem is that, given the relative simplicity of its formulation, finding the optimal solution (the optimal route). This problem is very difficult and belongs to the so-called class of NP-complete problems. Moreover, according to the existing classification, the traveling salesman problem is an example of an optimization problem that is an example of the most complex subclass of this class. However, the main subject of the paper is not the problem, but the method of its soluti- on, i.e. the branch and bound method. It consists of several related heuristics, and in the monographs, such a multi-heuristic branch and bound method was apparently not previously noted: the developers of algorithms and programs should have understood this themselves. At the same time, the method itself can be applied (with minor changes) to many other discrete optimization problems. So, the classical version of branch and bound method is the main subject of this paper, but also important is the second subject, i.e. the traveling salesman problem, also in the classical formulation. The paper deals with the application of the branch and bound method in solving the traveling salesman problem, and about this application, we can also use the word “classical”. However, in addition to the classic version of this implementation, we consider some new heuristics, related to the need to develop real-time algorithms.

scholarly journals A return to the traveling salesman model: a network branch and bound approach

2013 ◽  
Vol 16 (1) ◽  
pp. 52-63 ◽  
Author(s):  
Elias Munapo
Keyword(s):  
Linear Programming ◽  
Traveling Salesman Problem ◽  
Branch And Bound ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Traveling Salesman ◽  
Tree Model ◽  
The Traveling Salesman Problem ◽  
Assignment Models

This paper presents a network branch and bound approach for solving the traveling salesman problem. The problem is broken into sub-problems, each of which is solved as a minimum spanning tree model. This is easier to solve than either the linear programming-based or assignment models. 

scholarly journals Branch and Bound Algorithm for the Traveling Salesman Problem is not a Direct Type Algorithm

2020 ◽  
Vol 27 (1) ◽  
pp. 72-85
Author(s):  
Aleksandr N. Maksimenko
Keyword(s):  
Combinatorial Optimization ◽  
Traveling Salesman Problem ◽  
Branch And Bound ◽  
Optimization Problems ◽  
Clique Number ◽  
Traveling Salesman ◽  
Branch And Bound Algorithm ◽  
Combinatorial Optimization Problems ◽  
Type Algorithm ◽  
The Traveling Salesman Problem

In this paper, we consider the notion of a direct type algorithm introduced by V. A. Bondarenko in 1983. A direct type algorithm is a linear decision tree with some special properties. the concept of a direct type algorithm is determined using the graph of solutions of a combinatorial optimization problem. ‘e vertices of this graph are all feasible solutions of a problem. Two solutions are called adjacent if there are input data for which these and only these solutions are optimal. A key feature of direct type algorithms is that their complexity is bounded from below by the clique number of the solutions graph. In 2015-2018, there were five papers published, the main results of which are estimates of the clique numbers of polyhedron graphs associated with various combinatorial optimization problems. the main motivation in these works is the thesis that the class of direct type algorithms is wide and includes many classical combinatorial algorithms, including the branch and bound algorithm for the traveling salesman problem, proposed by J. D. C. Little, K. G. Murty, D. W. Sweeney, C. Karel in 1963. We show that this algorithm is not a direct type algorithm. Earlier, in 2014, the author of this paper showed that the Hungarian algorithm for the assignment problem is not a direct type algorithm. ‘us, the class of direct type algorithms is not so wide as previously assumed.

Research of Features of the Combined Algorithm for Solving the Asymmetric Traveling Salesman Problem

2021 ◽  
Vol 27 (1) ◽  
pp. 3-8
Author(s):  
M. V. Ulyanov ◽  
◽  
M. I. Fomichev ◽  
◽  
◽  
...  
Keyword(s):  
Decision Tree ◽  
Traveling Salesman Problem ◽  
Branch And Bound ◽  
Exact Algorithm ◽  
Traveling Salesman ◽  
Branch And Bound Method ◽  
Asymmetric Traveling Salesman Problem ◽  
The Individual ◽  
The Traveling Salesman Problem ◽  
Combined Algorithm

The exact algorithm that implements the Branch and Boimd method with precomputed tour which is calculated by Lin-Kernighan-Helsgaun metaheuristic algorithm for solving the Traveling Salesman Problem is concerned here. Reducing the number of decision tree nodes, which are created by the Branches and Bound method, due to a "good" precomputed tour leads to the classical balancing dilemma of time costs. A tour that is close to optimal one takes time, even when the Lin-Kernighan-Helsgaun algorithm is used, however it reduces the working time of the Branch and Bound method. The problem of determining the scope of such a combined algorithm arises. In this article it is solved by using a special characteristic of the individual Traveling Salesman Problem — the number of changes tracing direction in the search decision tree generated by the Branch and Bound Method. The use of this characteristic allowed to divide individual tasks into three categories, for which, based on experimental data, recommendations of the combined algorithm usage are formulated. Based on the data obtained in a computational experiment (in range from 30 to 45), it is recommended to use a combined algorithm for category III problems starting with n = 36, and for category II problems starting with n = 42.

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