POPMUSIC for the travelling salesman problem

POPMUSIC— Partial OPtimization Metaheuristic Under Special Intensification Conditions — is a template for tackling large problem instances. This metaheuristic has been shown to be very efficient for various hard combinatorial problems such as p-median, sum of squares clustering, vehicle routing, map labelling and location routing. A key point for treating large Travelling Salesman Problem (TSP) instances is to consider only a subset of edges connecting the cities. The main goal of this article is to present how to build a list of good candidate edges with a complexity lower than quadratic in the context of TSP instances given by a general function. The candidate edges are found with a technique exploiting tour merging and the POPMUSIC metaheuristic. When these candidate edges are provided to a good local search engine, high quality solutions can be found quite efficiently. The method is tested on TSP instances of up to several million cities with different structures (Euclidean uniform, clustered, 2D to 5D, grids, toroidal distances). Numerical results show that solutions of excellent quality can be obtained with an empirical complexity lower than quadratic without exploiting the geometrical properties of the instances.     

The seriation problem and the travelling salesman problem

The relationship between two combinatorial problems is explained in this paper. The first problem is the well known travelling salesman problem. The other problem, the seriation problem, is frequently encountered in archeology and has applications in genetics and in graph theory. Recent efforts to solve the seriation problem have produced numerous statistical and heuristic solutions. However algorithms that produce global optima remain very unsatisfactory. A new exact algorithm based on the interdependence between the travelling salesman problem and the seriation problem is presented here. It can handle larger problems than any of the existing exact algorithms.     

Maximum travelling salesman problem

We present different types of techniques for designing algorithms with worst-case performances for the Maximum Travelling Salesman Problem. Supported by Byelarussian Fundamental Science Found and DAAD     

Selective generalized travelling salesman problem

ABSTRACT

This paper introduces the Selective Generalized Traveling Salesman Problem (SGTSP). In SGTSP, the goal is to determine the maximum profitable tour within the given threshold of the tour’s duration, which consists of a subset of clusters and a subset of nodes in each cluster visited on the tour. This problem is a combination of cluster and node selection and determining the shortest path between the selected nodes. We propose eight mixed integer programming (MIP) formulations for SGTSP. All of the given MIP formulations are completely new, which is one of the major novelties of the study. The performance of the proposed formulations is evaluated on a set of test instances by conducting 4608 experimental runs. Overall, 4138 out of 4608 (~90%) test instances were solved optimally by using all formulations.     

A stochastic travelling salesman problem

The extended compound renewal process, a generalization of the concept of filtered Poisson process, is introduced. It is shown that its characteristic function is expressible as the solution of a second type Volterua integral equation is solved for some special cases. Moreover, the equation is used to find the first moment and a recursive relationship for the higher order raw moment of the process. Finally, several areas of applications to the GI^x/G/∞ queue are investigates including the size of the system, the queue output and the total backlog     

The travelling maintainer problem: integration of condition-based maintenance with the travelling salesman problem

The Travelling Salesman Problem (TSP) is one of the most studied problems in the literature due to its applicability to a large number of real cases. Most variants of the TSP consider total distance travelled. This paper presents a new generalised formulation of the TSP that aims to minimise the sum of functions of latencies to cities, rather than total distance travelled. Then, a new problem that uses a special function using the latency as input is presented, called the Travelling Maintainer Problem (TMP). The TMP integrates the output of prognostics in Condition-based Maintenance (CBM) with the TSP. CBM aims to minimise the failure and maintenance cost by identifying and predicting upcoming failures through the analysis of sensory information collected in real-time. Maintenance scheduling is performed using the predicted failure information obtained from the CBM. When the systems to be maintained are geographically distributed, maintenance scheduling requires integrated analysis of travel times and their effects on the failure progression in systems. This paper also presents Genetic Algorithm and Particle Swarm Optimisation-based solutions and their comparisons for the TMP on a case study.     

A self-organising model for the travelling salesman problem

This work describes a new algorithm, based on a self-organising neural network approach, to solve the Travelling Salesman Problem (TSP). Firstly, various features of the available adaptive neural network algorithms for TSP are reviewed and a new algorithm is proposed. In order to investigate the performance of the algorithms, a comprehensive empirical study has been provided. The simulations, which are conducted on a series of standard data, evaluate the overall performance of this approach by comparing the results with the best known or the optimal solutions of the problems. The proposed algorithm shows significant advances in both the quality of the solution and computational effort for most of the experimental data. The deviation from the optimal solution of this algorithm was, in the worst case, around 2%. This fact indicates that the self-organising neural network may be regarded as a promising heuristic approach for optimisation problems.     

Identification of non-optimal arcs for the travelling salesman problem

Conditions are presented for the identification of (directed) arcs for the traveling salesman problem, that can be eliminated with at least one optimal solution remaining. The conditions are not based on lower or upper bounds; the presence of an identified arc in a solution implies that the solution is not 3-optimal. An example illustrates how to use the conditions.     

A random-key genetic algorithm for the generalized traveling salesman problem

The generalized traveling salesman problem is a variation of the well-known traveling salesman problem in which the set of nodes is divided into clusters; the objective is to find a minimum-cost tour passing through one node from each cluster. We present an effective heuristic for this problem. The method combines a genetic algorithm (GA) with a local tour improvement heuristic. Solutions are encoded using random keys, which circumvent the feasibility problems encountered when using traditional GA encodings. On a set of 41 standard test problems with symmetric distances and up to 442 nodes, the heuristic found solutions that were optimal in most cases and were within 1% of optimality in all but the largest problems, with computation times generally within 10 seconds. The heuristic is competitive with other heuristics published to date in both solution quality and computation time.     

Guaranteed performance heuristics for the bottleneck travelling salesman problem

We consider constant-performance, polynomial-time, nonexact algorithms for the minimax or bottleneck version of the Travelling Salesman Problem. It is first shown that no such algorithm can exist for problems with arbitrary costs unless P = NP. However, when costs are positive and satisfy the triangle inequality, we use results pertaining to the squares of biconnected graphs to produce a polynomial-time algorithm with worst-case bound 2 and show further that, unless P = NP, no polynomial alternative can improve on this value.     

Tabu search for solving the black-and-white travelling salesman problem

The black-and-white travelling salesman problem (BWTSP) is an extension to the well-known TSP by partitioning the set of vertices into black and white vertices, and imposing cardinality and length constraints between two consecutive black vertices in a Hamiltonian tour. BWTSP has various applications in aircraft routing, telecommunication network design and logistics. In this paper, we develop several tabu search (TS) heuristics for solving the BWTSP. Our TS is built upon a new efficient neighbourhood structure, which exploits both the permutation and knapsack features of BWTSP. We also embed our TS as a heuristic procedure to improve the upper bound in a mixed-integer linear programming method. Extensive computational experiment on both benchmark and randomly generated instances shows effectiveness and efficiency of our algorithms. Our algorithms are able to obtain optimal and near optimal solutions to small instances in seconds, and find feasible solutions to large instances that have not been solved by the existing methods in the literature.     

A tour construction heuristic for the travelling salesman problem

The tour construction heuristic that generates initial tours for the tour improvement heuristics plays an important role in solving the travelling salesman problem (TSP). With the help of an effective tour construction heuristic, the performance of a heuristic can be improved. In this study we present a new tour construction algorithm, the construction priority (CP). We incorporate the performance of the CP into metaheuristics such as tabu search, genetic algorithm, space smoothing, and noising methods. The performance of the CP is empirically compared with the nearest neighbour (NN) approach. Extensive computational comparison shows that the CP is considerably more effective compared to NN.     

Some applications of the clustered travelling salesman problem

The purpose of this article is to describe several applications of the clustered travelling salesman problem arising in areas as diverse as vehicle routing, manufacturing, computer operations, examination timetabling, cytological testing, and integrated circuit testing.     

Integer programming approaches to the travelling salesman problem

The availability of an LP routine where we can add constraints and reoptimize, makes it possible to adopt an integer programming approach to the travelling-salesman problem.Starting with some of the constraints that define the problem we use either a branching process or a cutting planes routine to eliminate fractional solutions. We then test the resulting integer solution against feasibility and if necessary we generate the violated constraints and reoptimize until a genuine feasible solution is achieved.Usually only a small number of the omitted constraints is generated.The generality of the method and the modest solution times achieved leads us to believe that such an LP approach to other combinatorial problems deserves further consideration.     

The x-and-y-axes travelling salesman problem

The x-and-y-axes travelling salesman problem forms a special case of the Euclidean TSP, where all cities are situated on the x-axis and on the y-axis of an orthogonal coordinate system of the Euclidean plane. By carefully analyzing the underlying combinatorial and geometric structures, we show that this problem can be solved in polynomial time. The running time of the resulting algorithm is quadratic in the number of cities.     

Facets and valid inequalities for the time-dependent travelling salesman problem

The Time-Dependent Travelling Salesman Problem (TDTSP) is a generalization of the traditional TSP where the travel cost between two cities depends on the moment of the day the arc is travelled. In this paper, we focus on the case where the travel time between two cities depends not only on the distance between them, but also on the position of the arc in the tour. We consider two formulations proposed in the literature, we analyze the relationship between them and derive several families of valid inequalities and facets. In addition to their theoretical properties, they prove to be very effective in the context of a Branch and Cut algorithm.     

An additive bounding procedure for the asymmetric travelling salesman problem

In this paper, new lower bounds for the asymmetric travelling salesman problem are presented, based on spanning arborescences. The new bounds are combined in an additive procedure whose theoretical performance is compared with that of the Balas and Christofides procedure (1981). Both procedures have been imbedded in a simple branch and bound algorithm and experimentally evaluated on hard test problems.     

An improved branching rule for the symmetric travelling salesman problem

This paper presents an improvement to an existing branch and bound algorithm for solving the symmetric travelling salesman problem. The lower bound used is the standard one obtained from the sequence of minimal spanning 1-trees computed via subgradient optimization, but the branching rule is new. Rather than selecting for inclusion or exclusion those edges which violate the tour constraints in the final 1-tree of the sequence, we consider edges which are present in roughly half of the final few 1-trees. This results in a decision tree whose number of nodes grows by powers of two rather than three, hence significantly reducing the total number of decision nodes required to solve the problem.     

A domination algorithm for {0,1}‐instances of the travelling salesman problem

We present an approximation algorithm for ‐instances of the travelling salesman problem which performs well with respect to combinatorial dominance. More precisely, we give a polynomial‐time algorithm which has domination ratio . In other words, given a ‐edge‐weighting of the complete graph on vertices, our algorithm outputs a Hamilton cycle of with the following property: the proportion of Hamilton cycles of whose weight is smaller than that of is at most . Our analysis is based on a martingale approach. Previously, the best result in this direction was a polynomial‐time algorithm with domination ratio for arbitrary edge‐weights. We also prove a hardness result showing that, if the Exponential Time Hypothesis holds, there exists a constant such that cannot be replaced by in the result above. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 427–453, 2016