On the high multiplicity traveling salesman problem

This paper considers a version of the traveling salesman problem where the cities are to be visited multiple times. Each city has its own required number of visits. We investigate how the optimal solution and its objective value change when the numbers of visits are increased by a common multiplicator. In addition, we derive lower bounds on values of the multiplicator beyond which further increase does not improve the average tour length. Moreover, we show how and when the structure of an optimal tour length can be derived from tours with smaller multiplicities.     

Risky traveling salesman problem

In this paper we introduce a methodology for optimizing the expected cost of routing a single vehicle which has a probability of breaking down or failing to complete some of its tasks. More specifically, a calculus is devised for finding the optimal order in which each site should be visited.     

Submodularity and the traveling salesman problem

In this paper we investigate the relationship between traveling salesman tour lengths and submodular functions. This work is motivated by the one warehouse multi-retailer inventory/distribution problem with traveling salesman tour vehicle routing costs. Our goal is to find a submodular function whose values are close to those of optimal tour lengths through a central warehouse and a group of retailers. Our work shows that a submodular approximation to traveling salesman tour lengths whose error is bounded by a constant does not exist. However, we present heuristics that have errors which grow slowly with the number of retailers for the traveling salesman problem in the Euclidean plane. Furthermore, we perform computational tests that show that our submodular approximations of traveling salesman tour lengths have smaller errors than our theoretical worst case analysis would lead us to believe.     

Uncertain multiobjective traveling salesman problem

Traveling salesman problem is a fundamental combinatorial optimization model studied in the operations research community for nearly half a century, yet there is surprisingly little literature that addresses uncertainty and multiple objectives in it. A novel TSP variation, called uncertain multiobjective TSP (UMTSP) with uncertain variables on the arc, is proposed in this paper on the basis of uncertainty theory, and a new solution approach named uncertain approach is applied to obtain Pareto efficient route in UMTSP. Considering the uncertain and combinatorial nature of UMTSP, a new ABC algorithm inserted with reverse operator, crossover operator and mutation operator is designed to this problem, which outperforms other algorithms through the performance comparison on three benchmark TSPs. Finally, a new benchmark UMTSP case study is presented to illustrate the construction and solution of UMTSP, which shows that the optimal route in deterministic TSP can be a poor route in UMTSP.     

On the nearest neighbor rule for the traveling salesman problem

Rosenkrantz et al. (SIAM J. Comput. 6 (1977) 563) and Johnson and Papadimitriou (in: E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys (Eds.), The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley, Chichester, 1985, pp. 145-180, (Chapter 5)) constructed families of TSP instances with n cities for which the nearest neighbor rule yields a tour-length that is a factor above the length of the optimal tour.We describe two new families of TSP instances, for which the nearest neighbor rule shows the same bad behavior. The instances in the first family are graphical, and the instances in the second family are Euclidean. Our construction and our arguments are extremely simple and suitable for classroom use.     

The hierarchical traveling salesman problem

The distribution of relief aid is a complex problem where the operations have to be managed efficiently due to limited resources. We present a routing problem for relief operations whose primary goal is to satisfy demand for relief supplies at many locations taking into account the urgency of each demand. We have a single vehicle of unlimited capacity. Each node (location) has a demand and a priority. The priority indicates the urgency of the demand. Typically, nodes with the highest priorities need to be visited before lower priority nodes. We describe a new and interesting model for humanitarian relief routing that we call the hierarchical traveling salesman problem (HTSP). We compare the HTSP and the classical TSP in terms of worst-case behavior. We obtain a simple, but elegant result that exhibits the fundamental tradeoff between efficiency (distance) and priority and we provide several related observations and theorems.     

On the one-commodity pickup-and-delivery traveling salesman problem with stochastic demands

This paper studies how to set the vehicle capacity for traveling Salesman Problems where some of the customer demands are stochastic. The analyses are done for the one-commodity pickup-and-delivery TSP, as this problem also includes the setting of the initial load. The paper first considers feasibility issues. This includes finding the smallest vehicle capacity and some initial load such that a given tour is feasible for all scenarios. Different variants are considered as a function of the time when information becomes available. The paper then analyzes the case where some penalties are paid for routing a tour unable to handle customer demands. Various types of penalties are considered. The paper studies properties of the minimal expected penalty of a given tour, which are then used to provide approaches to find near-optimal tours. Computational results are presented.     

Genetic algorithms for the traveling salesman problem

This paper is a survey of genetic algorithms for the traveling salesman problem. Genetic algorithms are randomized search techniques that simulate some of the processes observed in natural evolution. In this paper, a simple genetic algorithm is introduced, and various extensions are presented to solve the traveling salesman problem. Computational results are also reported for both random and classical problems taken from the operations research literature.     

The on-line asymmetric traveling salesman problem

We consider two on-line versions of the asymmetric traveling salesman problem with triangle inequality. For the homing version, in which the salesman is required to return in the city where it started from, we give a $\frac{3+\sqrt{5}}{2}$-competitive algorithm and prove that this is best possible. For the nomadic version, the on-line analogue of the shortest asymmetric Hamiltonian path problem, we show that the competitive ratio of any on-line algorithm depends on the amount of asymmetry of the space in which the salesman moves. We also give bounds on the competitive ratio of on-line algorithms that are zealous, that is, in which the salesman cannot stay idle when some city can be served.     

Special cases of the traveling salesman problem

The paper surveys special cases of the traveling salesman problem that are in the literature. The tables include polynomial time recognition algorithms and relationships among special cases. As a result of research, the most general special cases are identified. The paper also contains a table of subclasses of the most general special cases. Applications and open questions are indicated.     

Recognition of Gilmore-Gomory traveling salesman problem

The best known special case of the traveling salesman problem that is well solved is that due to Gilmore and Gomory. It is a problem of scheduling a heat treatment furnace. In this paper we give a polynomial algorithm for detecting whether or not a given traveling salesman problem is an instance of their problem; we also compute the parameters necessary to apply their algorithm. Thus all such problems can not only be solved in polynomial time but also recognized in polynomial time.     

The symmetric quadratic traveling salesman problem

In the quadratic traveling salesman problem a cost is associated with any three nodes traversed in succession. This structure arises, e.g., if the succession of two edges represents energetic conformations, a change of direction or a possible change of transportation means. In the symmetric case, costs do not depend on the direction of traversal. We study the polyhedral structure of a linearized integer programming formulation of the symmetric quadratic traveling salesman problem. Our constructive approach for establishing the dimension of the underlying polyhedron is rather involved but offers a generic path towards proving facetness of several classes of valid inequalities. We establish relations to facets of the Boolean quadric polytope, exhibit new classes of polynomial time separable facet defining inequalities that exclude conflicting configurations of edges, and provide a generic strengthening approach for lifting valid inequalities of the usual traveling salesman problem to stronger valid inequalities for the symmetric quadratic traveling salesman problem. Applying this strengthening to subtour elimination constraints gives rise to facet defining inequalities, but finding a maximally violated inequality among these is NP-complete. For the simplest comb inequality with three teeth the strengthening is no longer sufficient to obtain a facet. Preliminary computational results indicate that the new cutting planes may help to considerably improve the quality of the root relaxation in some important applications.     

Provably good solutions for the traveling salesman problem

The determination of true optimum solutions of combinatorial optimization problems is seldomly required in practical applications. The majority of users of optimization software would be satisfied with solutions of guaranteed quality in the sense that it can be proven that the given solution is at most a few percent off an optimum solution. This paper presents a general framework for practical problem solving with emphasis on this aspect. A detailed discussion along with a report about extensive computational experiments is given for the traveling salesman problem.     

Continuous line drawings via the traveling salesman problem

We describe how to use the traveling salesman problem to create continuous line drawings of target pictures.     

Iterative patching and the asymmetric traveling salesman problem

Although Branch-and-Bound (BnB) methods are among the most widely used techniques for solving hard problems, it is still a challenge to make these methods smarter. In this paper, we investigate iterative patching, a technique in which a fixed patching procedure is applied at each node of the BnB search tree for the Asymmetric Traveling Salesman Problem. Computational experiments show that iterative patching results in general in search trees that are smaller than the classical BnB trees, and that solution times are lower for usual random and sparse instances. Furthermore, it turns out that, on average, iterative patching with the Contract-or-Patch procedure of Glover, Gutin, Yeo and Zverovich (2001) and the Karp–Steele procedure are the fastest, and that ‘iterative’ Modified Karp–Steele patching generates the smallest search trees.     

A solvable case of the traveling salesman problem

LetD = d _ij be the distance matrix defining a traveling salesman problem. IfD is upper triangular, i.e.d _ij = 0, fori j, then the traveling salesman problem can be solved with an amount of computation approximately equal to that required for an assignment problem of the same dimension.     

Certain generalization of the maximum traveling salesman problem

The problem is considered of finding in a complete undirected weighted graph a connected spanning subgraph with the given degrees of the vertices having maximum total weight of the edges. An approximate polynomial algorithm is presented for this problem. The algorithm is analyzed, and some upper bounds of its error are established in the general case as well as in the metric and Euclidean cases.     

The multi-commodity one-to-one pickup-and-delivery traveling salesman problem

This paper concerns a generalization of the traveling salesman problem (TSP) called multi-commodity one-to-one pickup-and-delivery traveling salesman problem (m-PDTSP) in which cities correspond to customers providing or requiring known amounts of m different commodities, and the vehicle has a given upper-limit capacity. Each commodity has exactly one origin and one destination, and the vehicle must visit each customer exactly once. The problem can also be defined as the capacitated version of the classical TSP with precedence constraints. This paper presents two mixed integer linear programming models, and describes a decomposition technique for each model to find the optimal solution. Computational experiments on instances from the literature and randomly generated compare the techniques and show the effectiveness of our implementation.