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.     

The precedence-constrained asymmetric traveling salesman polytope

Many applications of the traveling salesman problem require the introduction of additional constraints. One of the most frequently occurring classes of such constraints are those requiring that certain cities be visited before others (precedence constraints). In this paper we study the Precedence-Constrained Asymmetric Traveling Salesman (PCATS) polytope, i.e. the convex hull of incidence vectors of tours in a precedence-constrained directed graph. We derive several families of valid inequalities, and give polynomial time separation algorithms for important subfamilies. We then establish the dimension of the PCATS polytope and show that, under reasonable assumptions, the two main classes of inequalities derived are facet inducing.An early version of this paper was presented at the Oberwolfach Conference on Combinatorial Optimization in January 1991. This research was supported in part by the National Science Foundation, Grant #DDM-8901495 and the Office of Naval Research through Contract N00014-85-K-0198.Corresponding author.The work of this author was supported by MURST, Italy.     

New tighter polynomial length formulations for the asymmetric traveling salesman problem with and without precedence constraints

We propose a new formulation for the asymmetric traveling salesman problem, with and without precedence relationships, which employs a polynomial number of subtour elimination constraints that imply an exponential subset of certain relaxed Dantzig-Fulkerson-Johnson subtour constraints. Promising computational results are presented, particularly in the presence of precedence constraints.     

Genetic algorithm for asymmetric traveling salesman problem with imprecise travel times

This paper presents a variant of the asymmetric traveling salesman problem (ATSP) in which the traveling time between each pair of cities is represented by an interval of values (wherein the actual travel time is expected to lie) instead of a fixed (deterministic) value as in the classical ATSP. Here the ATSP (with interval objective) is formulated using the usual interval arithmetic. To solve the interval ATSP (I-ATSP), a genetic algorithm with interval valued fitness function is proposed. For this purpose, the existing revised definition of order relations between interval numbers for the case of pessimistic decision making is used. The proposed algorithm is based on a previously published work and includes some new features of the basic genetic operators. To analyze the performance and effectiveness of the proposed algorithm and different genetic operators, computational studies of the proposed algorithm on some randomly generated test problems are reported.     

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.     

A class of lifted path and flow-based formulations for the asymmetric traveling salesman problem with and without precedence constraints

In this paper, we present a new class of polynomial length formulations for the asymmetric traveling salesman problem (ATSP) by lifting an ordered path-based model using logical restrictions in concert with the Reformulation–Linearization Technique (RLT). We show that a relaxed version of this formulation is equivalent to a flow-based ATSP model, which in turn is tighter than the formulation based on the exponential number of Dantzig–Fulkerson–Johnson (DFJ) subtour elimination constraints. The proposed lifting idea is applied to derive a variety of new formulations for the ATSP, and we explore several dominance relationships among these. We also extend these formulations to include precedence constraints in order to enforce a partial order on the sequence of cities to be visited in a tour. Computational results are presented to exhibit the relative tightness of our formulations and the efficacy of the proposed lifting process.     

A parallel branch and bound algorithm for solving large asymmetric traveling salesman problems

A parallel branch and bound algorithm that solves the asymmetric traveling salesman problem to optimality is described. The algorithm uses an assignment problem based lower bounding technique, subtour elimination branching rules, and a subtour patching algorithm as an upper bounding procedure. The algorithm is organized around a data flow framework for parallel branch and bound. The algorithm begins by converting the cost matrix to a sparser version in such a fashion as to retain the optimality of the final solution. Computational results are presented for three different classes of problem instances: (1) matrix elements drawn from a uniform distribution of integers for instances of size 250 to 10 000 cities, (2) instances of size 250 to 1000 cities that concentrate small elements in the upper left portion of the cost matrix, and (3) instances of size 300 to 3000 cities that are designed to confound neighborhood search heuristics.     

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.     

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.     

A note on the prize collecting traveling salesman problem

We study the version of the prize collecting traveling salesman problem, where the objective is to find a tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. We present an approximation algorithm with constant bound. The algorithm is based on Christofides' algorithm for the traveling salesman problem as well as a method to round fractional solutions of a linear programming relaxation to integers, feasible for the original problem.Research supported in part by ONR contract N00014-90-J-1649 and NSF contract DDM-8922712.     

On patching algorithms for random asymmetric travelling salesman problems

Let the arc-lengthsL _ij of a complete digraph onn vertices be independent uniform [0, 1] random variables. We consider the patching algorithm of Karp and Steele for the travelling salesman problem on such a digraph and give modifications which tighten the expected error. We extend these ideas to thek-person travelling salesman problem and also consider the case where cities can be visited more than once.     

A hierarchical strategy for solving traveling salesman problems using elastic nets

In this article, we focus on implementing the elastic net method to solve the traveling salesman problem using a hierarchical approach. The result is a significant speed-up, which is studied both analytically and experimentally.     

Expanding neighborhood search–GRASP for the probabilistic traveling salesman problem

The Probabilistic Traveling Salesman Problem is a variation of the classic traveling salesman problem and one of the most significant stochastic routing problems. In probabilistic traveling salesman problem only a subset of potential customers need to be visited on any given instance of the problem. The number of customers to be visited each time is a random variable. In this paper, a variant of the well-known Greedy Randomized Adaptive Search Procedure (GRASP), the Expanding Neighborhood Search–GRASP, is proposed for the solution of the probabilistic traveling salesman problem. expanding neighborhood search–GRASP has been proved to be a very efficient algorithm for the solution of the traveling salesman problem. The proposed algorithm is tested on a numerous benchmark problems from TSPLIB with very satisfactory results. Comparisons with the classic GRASP algorithm and with a Tabu Search algorithm are also presented. Also, a comparison is performed with the results of a number of implementations of the Ant Colony Optimization algorithm from the literature and in six out of ten cases the proposed algorithm gives a new best solution.     

NP-completeness of the hamming salesman problem

It is shown that the traveling salesman problem, where cities are bit strings with Hamming distances, is NP-complete.     

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.     

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.     

A tabu search heuristic using genetic diversification for the clustered traveling salesman problem

The clustered traveling salesman problem is an extension of the classical traveling salesman problem where the set of vertices is partitioned into clusters. The objective is to find a least cost Hamiltonian cycle such that the vertices of each cluster are visited contiguously and the clusters are visited in a prespecified order. A tabu search heuristic is proposed to solve this problem. This algorithm periodically restarts its search by merging two elite solutions to form a new starting solution (in a manner reminiscent of genetic algorithms). Computational results are reported on sets of Euclidean problems with different characteristics.     

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.