Symmetric Traveling Salesman Problem Research Articles

Fictitious upper bounds in an algorithm for the symmetric traveling salesman problem

The performance of a branch and bound algorithm depends on the quality of the upper bounds. For a depth first search strategy this dependence is larger than for a breadth first strategy. If no good upper bounding heuristic is available, one may decide to use fictitious upper bounds. We discuss the use of three types of these bounds in an algorithm for the symmetric traveling salesman problem (TSP). Computational results illustrate the discussion for problems with up to 200 cities and of different problem types.

New lower bounds for the Symmetric Travelling Salesman Problem

In this paper new lower bounds for the Symmetric Travelling Salesman Problem are proposed and combined in additive bounding procedures. Efficient implementations of the algorithms are given; in particular, fast procedures for computing the linear programming reduced costs of the Shortest Spanning Tree (SST) Problem and for finding all ther-SST of a given graph, are presented. Computational results on randomly generated test problems are reported.

Constrained spanning trees and the traveling salesman problem

Minimum weight 1-trees provide a well-known lower bound for the symmetric traveling salesman problem. We propose to strengthen this bound by imposing degree-constraints upon the 1-trees. The vertices for the constraints are chosen to form a stable set S. We propose an O( m log log n + | S|( n| S| + mα( m, n))) algorithm for this problem and report on its use in a Lagrangian approach to the traveling salesman problem.

Classifying traveling salesman problems

When should two traveling salesman problems be equivalent? This paper contains an answer which is motivated by an example. The equivalence relation is applied to 0,1 symmetric traveling salesman problems. Open questions are indicated.

A polynomial-time solution to Papadimitriou and Steiglitz's ‘traps’

Papadimitriou and Steiglitz constructed ‘traps’ for the symmetric travelling salesman problem (TSP) with n = 8 k cities. The constructed problem instances have exponentially many suboptimal solutions with arbitrarily large weight, which differ from the unique optimal solution in exactly 3 k edges, and hence local search algorithms are ineffective to solve this problem. However, we show that this class of ‘catastrophic’ examples can be solved by linear programming relaxation appended with k subtour elimination constraints. It follows that this class of problem instances of TSP can be optimized in polynomial time.

Technical Note—An Improved Transformation of the Symmetric Multiple Traveling Salesman Problem

We improve the standard transformation of the symmetric, single-depot, multiple traveling salesman problem (MTSP) to one on a sparser edge configuration. The improvement tends to suppress, to a large extent, the degeneracy of the resulting symmetric traveling salesman problem (TSP). As a result, a 1-tree based TSP algorithm solves large MTSPs of the type we consider as easily as it solves standard TSPs. We demonstrate the improvement by presenting computational results that are currently the best available.

A new class of cutting planes for the symmetric travelling salesman problem

A comprehensive class of cutting planes for the symmetric travelling salesman problem (TSP) is proposed which contains the known “comb inequalities”, the “path inequalities” and the “3-star constraints” as special cases. Its relation to the “clique tree inequalities” is discussed. The cutting planes are shown to be valid for a relaxed version of the TSP, the travelling salesman problem on a road network, and—under certain conditions—to define facets of the polyhedron associated with this problem.

Optimization of a 532-city symmetric traveling salesman problem by branch and cut

We report the solution to optimality of a 532-city symmetric traveling salesman problem involving the optimization over 141,246 zero-one variables. The results of an earlier study by Crowder and Padberg [1] are cross-validated. In this note we briefly outline the methodology, algorithms and software system that we developed to obtain these results.

Accelerated branch exchange heuristics for symmetric traveling salesman problems

AbstractA method for accelerating the computational performance of branch exchange heuristics for symmetric traveling salesman problems (TSP's) is presented. The improvement in performance is obtained by considering only exchanges that have a good chance of producing a better solution. In the instance of the 3–optimal heuristic, the approach reduces the number of operations required to obtain a good solution to a TSP with N nodes from O(N3) to O(N2), without a corresponding degradation in the quality of the solution. Most of the results are produced for Euclidean TSP's, but evidence is presented that indicates that thes results apply equally well if not more strongly to the general symmetric TSP.

Clique Tree Inequalities and the Symmetric Travelling Salesman Problem

The linear programming cutting plane approach for solving the travelling salesman problem has recently proven to be highly successful, cf. Crowder and Padberg (Crowder, H. P., M. W. Padberg. 1980. Solving large-scale symmetric travelling salesman problems to optimality. Management Sci. 26 495–509.), Grötschel (Grötschel, M. 1980a. On the symmetric travelling salesman problem: Solution of a 120 city problem. Math. Programming Stud. 12 61–77.), Padberg and Hong (Padberg, M. W., S. Hong. 1980. On the symmetric travelling salesman problem: A computational study. Math. Programming Stud. 12 78–107.). One of the reasons for this success is certainly the fact that instead of ordinary cutting planes (Gomory-cuts etc.) problem-specific cutting planes could be used which define facets of the underlying integer programming polytopes. In this paper we shall define a new class of inequalities (clique tree inequalities) valid for the travelling salesman polytope which properly contains many of the known classes of inequalities (like subtour elimination constraints, 2-matching constraints, comb inequalities), and we show that all these new inequalities induce facets of the travelling salesman polytope. Since the general structure of these new inequalities is quite simple we hope that it will be possible to use the inequalities efficiently in cutting plane procedures for the travelling salesman problem.

Transforming asymmetric into symmetric traveling salesman problems: erratum

Our previously presented transformation of partly asymmetric into symmetric traveling salesman problems is shown to be incorrect. It only yields a lower bound on the optimal cost. We also show that asymmetric traveling salesman problems on n nodes can be solved as symmetric problems on 2 n − 1 nodes.

A cutting plane procedure for the travelling salesman problem on road networks

A cutting plane algorithm for the exact solution of the symmetric travelling salesman problem (TSP) is proposed. The real tours on a usually incomplete road network, which are in general non-Hamiltonian, are characterized directly by an integer linear programming model. The algorithm generates special cutting planes for this model. Computational results for real road networks with up to 292 visiting places are reported, as well as for classical problems of the literature with up to 120 cities. Some of the latter problems have been solved for the first time with a pure cutting plane approach.

A useful transform of standard input data for a classical NP-complete problem

The archetypal symmetric travelling salesman problem can be seen in a new and interesting way, by using first a standard preparatory phase of input data, and then by applying a transform from the set D of ‘distances’ among ‘cities’ and the set B of ‘loss of optimality’.The specific form of D→B transform is introduced and discussed. In order to show in realistic terms the interest of the approach proposed, a class of ‘diffusive’ heuristic procedures operating from B is defined.An example of solution by an algorithm included in this class is completely worked out; an outline of computational tests done on the same algorithm is also given.

On recursive computation of minimum spanning trees for special partial graphs

Let G = (E N) be an undirected graph and T be a minimum spanning tree. If one edge e of T is removed from G then a new minimum spanning tree Te can be obtained from T recursively. In this paper an algorithm is proposed to determine n— 1 =card (T) new minimum spanning trees Te , one for each edge e, of T. with a total expense of O (card (N)2) computational operations. In addition an application to the symmetric traveling salesman problem is given.

Nonoptimal Edges for the Symmetric Traveling Salesman Problem

For the symmetric traveling salesman problem, we identify a set of (undirected) edges that can be eliminated while still retaining at least one optimal solution of the problem. The simple identification rules are based on the fact that a solution can be optimal only if it is 2-optimal. The rules are less stringent than those formulated for nonoptimal arcs (directed), applied to the symmetric case. So, in general, they identify more nonoptimal edges. Application of the theory in a 1-tree based traveling salesman algorithm roughly halves the average computation time for Euclidean problems. We indicate to what extent the theorems presented here can be adapted for variants of the traveling salesman problem.

Worst-case analysis of two travelling salesman heuristics

In this paper we analyze the worst-case performance of some heuristics for the symmetric travelling salesman problem. We show that the worst-case ratios of tour length produced by the savings and greedy heuristics to that of a minimum tour are bounded by [log 2 n]+1 and 0.5([log 2 n]+1) respectively, where n is the number of cities.

Transforming asymmetric into symmetric traveling salesman problems

We describe how to transform an asymmetric traveling salesman problem into a symmetric one at the cost of almost doubled problem size. Use and consequences are discussed shortly.

The symmetric traveling salesman problem and edge exchanges in minimal 1-trees

In this article the effect of exchanging edges inside a minimal 1-tree with edges outside is analysed. In combination with an upper bound this analysis enables the elimination of variables in the symmetric traveling salesman problem. After discussion on a number of improvements for this analysis, the implementation is described in a traveling salesman algorithm based on the 1-tree relaxation. Computational results show the advantages of the edges exchanges for Euclidean problems (up to 120 cities) as well as for random table problems (up to 200 cities).

Solving symmetric vehicle routing problems asymmetrically

The vehicle routing problem can be regarded as a traveling salesman problem with additional constraints. Algorithms based on assignment relaxations provide better solutions for the symmetric traveling salesman problem if they are used on an asymmetric transformed distance matrix, as shown by Jonker et al., in a paper in Operations Research. The generalization of such a transformation to the distance matrix of symmetric vehicle routing problems is described. The approach is illustrated within a heuristic algorithm, although it can also be the basis for an exact algorithm. For a number of standard problems computational results are given, that are competitive to results of other algorithms.

A branch and bound algorithm for the symmetric traveling salesman problem based on the 1-tree relaxation

In 1970 Held and Karp introduced the Lagrangean approach to the symmetric traveling salesman problem. We use this 1-tree relaxation in a new branch and bound algorithm. It differs from other algorithms not only in the branching scheme, but also in the ascent method to calculate the 1-tree bounds. urthermore we determine heuristic solutions throughout the computations to provide upperbounds. We present computational results for both a depth-first and a breadth-first version of our algorithm. On the average our results on a number of Euclidean problems from the literature are obtained in about 60% less 1-trees than the best known algorithm based on the 1-tree relaxation. For random table problems (up to 100 cities) the average results are also satisfactory.