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Journal ArticleDOI

The Traveling-Salesman Problem and Minimum Spanning Trees

01 Dec 1970-Operations Research (INFORMS)-Vol. 18, Iss: 6, pp 1138-1162
TL;DR: It is shown that maxπwπ = C* precisely when a certain well-known linear program has an optimal solution in integers.
Abstract: This paper explores new approaches to the symmetric traveling-salesman problem in which 1-trees, which are a slight variant of spanning trees, play an essential role. A 1-tree is a tree together with an additional vertex connected to the tree by two edges. We observe that i a tour is precisely a 1-tree in which each vertex has degree 2, ii a minimum 1-tree is easy to compute, and iii the transformation on "intercity distances" cij → Cij + πi + πj leaves the traveling-salesman problem invariant but changes the minimum 1-tree. Using these observations, we define an infinite family of lower bounds wπ on C*, the cost of an optimum tour. We show that maxπwπ = C* precisely when a certain well-known linear program has an optimal solution in integers. We give a column-generation method and an ascent method for computing maxπwπ, and construct a branch-and-bound method in which the lower bounds wπ control the search for an optimum tour.
Citations
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Book
01 Jan 1976
TL;DR: In this paper, the authors present Graph Theory with Applications: Graph theory with applications, a collection of applications of graph theory in the field of Operational Research and Management. Journal of the Operational research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
Abstract: (1977). Graph Theory with Applications. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.

7,497 citations

Journal ArticleDOI
S. Lin1, Brian W. Kernighan1
TL;DR: This paper discusses a highly effective heuristic procedure for generating optimum and near-optimum solutions for the symmetric traveling-salesman problem based on a general approach to heuristics that is believed to have wide applicability in combinatorial optimization problems.
Abstract: This paper discusses a highly effective heuristic procedure for generating optimum and near-optimum solutions for the symmetric traveling-salesman problem. The procedure is based on a general approach to heuristics that is believed to have wide applicability in combinatorial optimization problems. The procedure produces optimum solutions for all problems tested, "classical" problems appearing in the literature, as well as randomly generated test problems, up to 110 cities. Run times grow approximately as n2; in absolute terms, a typical 100-city problem requires less than 25 seconds for one case GE635, and about three minutes to obtain the optimum with above 95 per cent confidence.

3,761 citations

Book
22 Jun 2009
TL;DR: This book provides a complete background on metaheuristics and shows readers how to design and implement efficient algorithms to solve complex optimization problems across a diverse range of applications, from networking and bioinformatics to engineering design, routing, and scheduling.
Abstract: A unified view of metaheuristics This book provides a complete background on metaheuristics and shows readers how to design and implement efficient algorithms to solve complex optimization problems across a diverse range of applications, from networking and bioinformatics to engineering design, routing, and scheduling. It presents the main design questions for all families of metaheuristics and clearly illustrates how to implement the algorithms under a software framework to reuse both the design and code. Throughout the book, the key search components of metaheuristics are considered as a toolbox for: Designing efficient metaheuristics (e.g. local search, tabu search, simulated annealing, evolutionary algorithms, particle swarm optimization, scatter search, ant colonies, bee colonies, artificial immune systems) for optimization problems Designing efficient metaheuristics for multi-objective optimization problems Designing hybrid, parallel, and distributed metaheuristics Implementing metaheuristics on sequential and parallel machines Using many case studies and treating design and implementation independently, this book gives readers the skills necessary to solve large-scale optimization problems quickly and efficiently. It is a valuable reference for practicing engineers and researchers from diverse areas dealing with optimization or machine learning; and graduate students in computer science, operations research, control, engineering, business and management, and applied mathematics.

2,735 citations


Cites background from "The Traveling-Salesman Problem and ..."

  • ...The Held–Karp (HK) 1-tree lower bound for the symmetric TSP problem is quick and easy to compute [371]....

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Journal ArticleDOI
TL;DR: This paper is a review of Lagrangian relaxation based on what has been learned in the last decade and has led to dramatically improved algorithms for a number of important problems in the areas of routing, location, scheduling, assignment and set covering.
Abstract: (This article originally appeared in Management Science, January 1981, Volume 27, Number 1, pp. 1-18, published by The Institute of Management Sciences.) One of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints. Dualizing the side constraints produces a Lagrangian problem that is easy to solve and whose optimal value is a lower bound (for minimization problems) on the optimal value of the original problem. The Lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm. This approach has led to dramatically improved algorithms for a number of important problems in the areas of routing, location, scheduling, assignment and set covering. This paper is a review of Lagrangian relaxation based on what has been learned in the last decade.

2,318 citations


Cites background or methods from "The Traveling-Salesman Problem and ..."

  • ...Held and Karp [27] experimented with primitive-direction ascent in their early work on the traveling salesman problem....

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  • ...It is also worth noting that many other successful Lagrangian relaxations (including Held and Karp [27], [28], Etcheberry [12], Etcheberry, et al....

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  • ...TRAVELING SALESMAN Symmetric Held & Karp [27], [28] Spanning Tree Helbig Hansen and Krarup [26] Spanning Tree Asymmetric Bazarra & Goode [3] Spanning Tree Symmetric Balas & Christofides [1] Perfect 2-Matching Asymmetric Balas & Christofides [I] Assignment...

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  • ...However, the "birth" of the Lagrangian approach as it exists today occurred in 1970 when Held and Karp [27], [28] used a Lagrangian problem based on minimum spanning trees to devise a dramatically successful algorithm for the traveling salesman problem....

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Journal ArticleDOI
TL;DR: In this paper, column generation methods for integer programs with a huge number of variables are discussed, including implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree.
Abstract: We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. We then discuss computational issues and implementation of column generation, branch-and-bound algorithms, including special branching rules and efficient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality.

2,248 citations

References
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Journal ArticleDOI
TL;DR: A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to

22,704 citations


"The Traveling-Salesman Problem and ..." refers background in this paper

  • ...Whereas the traveling-salesman problem is considered very difficult, the minimum spanning-tree problem can be solved by an algorithm that inspects each edge of K, exactly once[4,12] (see also section 6)....

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Journal ArticleDOI
01 Feb 1956
TL;DR: Kurosh and Levitzki as discussed by the authors, on the radical of a general ring and three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings.
Abstract: 7 A Kurosh, Ringtheoretische Probleme die mit dem Burnsideschen Problem uber periodische Gruppen in Zussammenhang stehen, Bull Acad Sei URSS, Ser Math vol 5 (1941) pp 233-240 8 J Levitzki, On the radical of a general ring, Bull Amer Math Soc vol 49 (1943) pp 462^66 9 -, On three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings, Trans Amer Math Soc vol 74 (1953) pp 384-409

5,104 citations


"The Traveling-Salesman Problem and ..." refers background in this paper

  • ...Whereas the traveling-salesman problem is considered very difficult, the minimum spanning-tree problem can be solved by an algorithm that inspects each edge of K, exactly once[4,12] (see also section 6)....

    [...]

Journal ArticleDOI
P. C. Gilmore1, Ralph E. Gomory1
TL;DR: In this paper, a technique is described for overcoming the difficulty in the linear programming formulation of the cutting-stock problem, which enables one to compute always with a matrix which has no more columns than it has rows.
Abstract: The cutting-stock problem is the problem of filling an order at minimum cost for specified numbers of lengths of material to be cut from given stock lengths of given cost. When expressed as an integer programming problem the large number of variables involved generally makes computation infeasible. This same difficulty persists when only an approximate solution is being sought by linear programming. In this paper, a technique is described for overcoming the difficulty in the linear programming formulation of the problem. The technique enables one to compute always with a matrix which has no more columns than it has rows.

1,933 citations

Book ChapterDOI
TL;DR: The RAND Corporation in the early 1950s contained Arrow, Bellman, Dantzig, Flood, Ford, Fulkerson, Gale, Johnson, Nash, Orchard-Hays, Robinson, Shapley, Simon, Wagner, and other household names as discussed by the authors.
Abstract: The RAND Corporation in the early 1950s contained “what may have been the most remarkable group of mathematicians working on optimization ever assembled” [6]: Arrow, Bellman, Dantzig, Flood, Ford, Fulkerson, Gale, Johnson, Nash, Orchard-Hays, Robinson, Shapley, Simon, Wagner, and other household names. Groups like this need their challenges. One of them appears to have been the traveling salesman problem (TSP) and particularly its instance of finding a shortest route through Washington, DC, and the 48 states [4, 7].

1,461 citations

Journal ArticleDOI
TL;DR: In this paper, a dynamic programming approach to the solution of three sequencing problems, namely, a scheduling problem involving arbitrary cost functions, the traveling-salesman problem, and an assembly line balancing problem, is presented.
Abstract: This paper explores a dynamic programming approach to the solution of three sequencing problems: a scheduling problem involving arbitrary cost functions, the traveling-salesman problem, and an assembly line balancing problem. Each of the problems is shown to admit of numerical solution through the use of a simple recursion scheme; these recursion schemes also exhibit similarities and contrasts in the structures of the three problems. For large problems, direct solution by means of dynamic programming is not practical, but procedures are given for obtaining good approximate results by solving sequences of smaller derived problems. Experience with a computer program for the solution of traveling-salesman problems is presented.

1,073 citations