3-Approximation algorithm for a two depot, heterogeneous traveling salesman problem

We present the first approximation algorithm for a two depot, heterogeneous traveling salesman problem with an approximation ratio of 3 when the costs are symmetric and satisfy the triangle inequality.     

A new extension principle algorithm for the traveling salesman problem

In this paper, an algorithm for solving the asymmetric traveling salesman problem is developed and tested by computation. This algorithm is based on the extension principle by Schoch and uses the assignment problem relaxation of the traveling salesman problem for computing lower bounds. Computational experience with randomly generated test problems indicate that the present algorithm yields good results in runtime which are comparable with the results of Smith/Srinivasan/Thompson. Computational experience are reported for up to 120-node problems with uniformly distributed and approximately normally distributed cost.     

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.     

随机容错设施选址问题的原始-对偶近似算法

研究两阶段随机容错设施选址问题,其中需要服务的顾客在第二阶段出现（在第一阶段不知道）．两个阶段中每个设施的开设费用可以不同,设施的开设依赖于阶段和需要服务的顾客集合（称为场景）．并且在出现的场景里的每个顾客都有相同的连接需求,即每个顾客需要由r个不同的设施服务．给定所有可能的场景及相应的概率,目标是在两个阶段分别选取开设的设施集合,将出现场景的顾客连接到r个不同的开设设施上,使得包括设施费用和连接费用的总平均费用最小．根据问题的特定结构,给出了原始。对偶（组合）3-近似算法．     

Probabilistic analysis of an approximation algorithm for the traveling salesman problem on unbounded above instances

The probabilistic analysis of a modification of the approximation algorithm “Nearest City” is presented for the traveling salesman minimization problem. The case is considered when the entries of the distance matrix are some independent identically distributed random variables with the values in the range [a _n ,∞), a _n > 0, that are unbounded above and have a truncated normal or exponential distribution. We obtain the estimates of relative errors, the failure probabilities, and also the conditions of asymptotic optimality of the algorithm. The modification of the algorithm allows us to perform analysis so that the obtained results are true for the traveling salesman problems in the cases of both directed and undirected graphs.     

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.     

An approximation algorithm for the traveling tournament problem

This paper describes the traveling tournament problem, a well-known benchmark problem in the field of tournament timetabling. We propose a new lower bound for the traveling tournament problem, and construct a randomized approximation algorithm yielding a feasible solution whose approximation ratio is less than 2+(9/4)/(n−1), where n is the number of teams. Additionally, we propose a deterministic approximation algorithm with the same approximation ratio using a derandomization technique. For the traveling tournament problem, the proposed algorithms are the first approximation algorithms with a constant approximation ratio, which is less than 2+3/4.     

A restricted dynamic programming heuristic algorithm for the time dependent traveling salesman problem

Dynamic programming (DP) algorithms for the traveling salesman problem (TSP) can easily incorporate time dependent travel times, time windows, and precedence relationships which present difficulties for algorithms based on linear or nonlinear programming formulations and for many TSP heuristics. However, exact DP algorithms for the TSP have exponential storage and computational time requirements and can solve only very small problems. We present a restricted DP heuristic (a generalization of the nearest neighbor heuristic) that can include all the above considerations but solves much larger problems. The heuristic cannot guarantee optimality because only a userspecified specified number of partial tours is retained at each stage. In this paper, the heuristic is implemented for the time dependent traveling salesman problem and is tested on a personal computer on randomly generated problems. The quality of the solution improves, on average, as more computational time is permitted.     

A primal-dual approximation algorithm for stochastic facility location problem with service installation costs

We consider the stochastic version of the facility location problem with service installation costs. Using the primal-dual technique, we obtain a 7-approximation algorithm.     

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.     

Differential approximation results for the traveling salesman problem with distances 1 and 2

We prove that both minimum and maximum traveling salesman problems on complete graphs with edge-distances 1 and 2 (denoted by min_TSP12 and max_TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard-approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any ϵ>0, it is NP-hard to approximate both problems better than within 741/742+ϵ. The same results hold when dealing with a generalization of min_ and max_TSP12, where instead of 1 and 2, edges are valued by a and b.     

The time dependent traveling salesman problem: polyhedra and algorithm

The time dependent traveling salesman problem (TDTSP) is a generalization of the classical traveling salesman problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 100 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet-defining cuts. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving almost all instances from the TSPLIB with up to 107 vertices.     

A note on heuristics for the traveling salesman problem

We introduce subclasses of the traveling salesman problem (TSP) and analyze the behaviour of heuristics on them. We derive bounds for the performance of the algorithms.     

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.     

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.     

An efficient genetic algorithm for the traveling salesman problem with precedence constraints

The traveling salesman problem with precedence constraints (TSPPC) is one of the most difficult combinatorial optimization problems. In this paper, an efficient genetic algorithm (GA) to solve the TSPPC is presented. The key concept of the proposed GA is a topological sort (TS), which is defined as an ordering of vertices in a directed graph. Also, a new crossover operation is developed for the proposed GA. The results of numerical experiments show that the proposed GA produces an optimal solution and shows superior performance compared to the traditional algorithms.     

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.     

An algorithm for the traveling salesman problem with pickup and delivery customers

The paper extends the branch and bound algorithm of Little, Murty, Sweeney, and Karel to the traveling salesman problem with pickup and delivery customers, where each pickup customer is required to be visited before its associated delivery customer. The problems considered include single and multiple vehicle cases as well as infinite and finite capacity cases. Computational results are reported.