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.     

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 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.     

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.     

A 2-approximation algorithm for the network substitution problem

The network substitution problem is to substitute an existing network for a new network so that to minimize the cost of exploiting the existing network during the period when the new network is being constructed. We show that this problem is NP-hard, and propose a 2-approximation algorithm for solving it.     

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.     

A 2.75-approximation algorithm for the unconstrained traveling tournament problem

A 2.75-approximation algorithm is proposed for the unconstrained traveling tournament problem, which is a variant of the traveling tournament problem. For the unconstrained traveling tournament problem, this is the first proposal of an approximation algorithm with a constant approximation ratio. In addition, the proposed algorithm yields a solution that meets both the no-repeater and mirrored constraints. Computational experiments show that the algorithm generates solutions of good quality.     

Adaptive sample size and importance sampling in estimation-based local search for the probabilistic traveling salesman problem

The probabilistic traveling salesman problem is a paradigmatic example of a stochastic combinatorial optimization problem. For this problem, recently an estimation-based local search algorithm using delta evaluation has been proposed. In this paper, we adopt two well-known variance reduction procedures in the estimation-based local search algorithm: the first is an adaptive sampling procedure that selects the appropriate size of the sample to be used in Monte Carlo evaluation; the second is a procedure that adopts importance sampling to reduce the variance involved in the cost estimation. We investigate several possible strategies for applying these procedures to the given problem and we identify the most effective one. Experimental results show that a particular heuristic customization of the two procedures increases significantly the effectiveness of the estimation-based local search.     

Efficient local search algorithms for known and new neighborhoods for the generalized traveling salesman problem

The generalized traveling salesman problem (GTSP) is a well-known combinatorial optimization problem with a host of applications. It is an extension of the Traveling Salesman Problem (TSP) where the set of cities is partitioned into so-called clusters, and the salesman has to visit every cluster exactly once.     

Extension of the 2-p-opt and 1-shift algorithms to the heterogeneous probabilistic traveling salesman problem

The probabilistic traveling salesman problem is a well known problem that is quite challenging to solve. It involves finding the tour with the lowest expected cost for customers that will require a visit with a given probability. There are several proposed algorithms for the homogeneous version of the problem, where all customers have identical probability of being realized. From the literature, the most successful approaches involve local search procedures, with the most famous being the 2-p-opt and 1-shift procedures proposed by Bertsimas [D.J. Bertsimas, L. Howell, Further results on the probabilistic traveling salesman problem, European Journal of Operational Research 65 (1) (1993) 68–95]. Recently, however, evidence has emerged that indicates the equations offered for these procedures are not correct, and even when corrected, the translation to the heterogeneous version of the problem is not simple. In this paper we extend the analysis and correction to the heterogeneous case. We derive new expressions for computing the cost of 2-p-opt and 1-shift local search moves, and we show that the neighborhood of a solution may be explored in O(n²) time, the same as for the homogeneous case, instead of O(n³) as first reported in the literature.     

A threshold accepting heuristic with intense local search for the solution of special instances of the traveling salesman problem

In real life scheduling, variations of the standard traveling salesman problem are very often encountered. The aim of this work is to present a new heuristic method for solving three such special instances with a common approach. The proposed algorithm uses a variant of the threshold accepting method, enhanced with intense local search, while the candidate solutions are produced through an insertion heuristic scheme. The main characteristic of the algorithm is that it does not require modifications and parameter tuning in order to cope with the three different problems. Computational results on a variety of real life and artificial problems are presented at the end of this work and prove the efficiency and the ascendancy of the proposed method over other algorithms found in the literature.     

An iterative rounding 2-approximation algorithm for the k-partial vertex cover problem

We study a generalization of the vertex cover problem. For a given graph with weights on the vertices and an integer k, we aim to find a subset of the vertices with minimum total weight, so that at least k edges in the graph are covered. The problem is called the k-partial vertex cover problem. There are some 2-approximation algorithms for the problem. In the paper we do not improve on the approximation ratios of the previous algorithms, but we derive an iterative rounding algorithm. We present our technique in two algorithms. The first is an iterative rounding algorithm and gives a （2 ＋ Q/OPT ）-approximation for the k-partial vertex cover problem where Q is the largest finite weight in the problem definition and OPT is the optimal value for the instance. The second algorithm uses the first as a subroutine and achieves an approximation ratio of 2.     

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.     

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.     

Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm

We present a Monte Carlo algorithm to find approximate solutions of the traveling salesman problem. The algorithm generates randomly the permutations of the stations of the traveling salesman trip, with probability depending on the length of the corresponding route. Reasoning by analogy with statistical thermodynamics, we use the probability given by the Boltzmann-Gibbs distribution. Surprisingly enough, using this simple algorithm, one can get very close to the optimal solution of the problem or even find the true optimum. We demonstrate this on several examples.We conjecture that the analogy with thermodynamics can offer a new insight into optimization problems and can suggest efficient algorithms for solving them.The author acknowledges stimulating discussions with J. Piút concerning the main ideas of the present paper. The author is also indebted to P. Brunovský, J. erný, M. Hamala, . Peko, . Znám, and R. Zajac for useful comments.     

A faster 2-approximation algorithm for the minmax p-traveling salesmen problem on a tree

Given an edge-weighted tree T and an integer p1, the minmax p-traveling salesmen problem on a tree T asks to find p tours such that the union of the p tours covers all the vertices. The objective is to minimize the maximum of length of the p tours. It is known that the problem is NP-hard and has a (2−2/(p+1))-approximation algorithm which runs in O(p^p−1n^p−1) time for a tree with n vertices. In this paper, we consider an extension of the problem in which the set of vertices to be covered now can be chosen as a subset S of vertices and weights to process vertices in S are also introduced in the tour length. For the problem, we give an approximation algorithm that has the same performance guarantee, but runs in O((p−1)!·n) time.