Asymmetric Traveling Salesman Problem Instances Research Articles

Adjustable Pheromone Reinforcement Strategies for Problems with Efficient Heuristic Information

Ant colony optimization (ACO) is a well-known class of swarm intelligence algorithms suitable for solving many NP-hard problems. An important component of such algorithms is a record of pheromone trails that reflect colonies’ experiences with previously constructed solutions of the problem instance that is being solved. By using pheromones, the algorithm builds a probabilistic model that is exploited for constructing new and, hopefully, better solutions. Traditionally, there are two different strategies for updating pheromone trails. The best-so-far strategy (global best) is rather greedy and can cause a too-fast convergence of the algorithm toward some suboptimal solutions. The other strategy is named iteration best and it promotes exploration and slower convergence, which is sometimes too slow and lacks focus. To allow better adaptability of ant colony optimization algorithms we use κ-best, max-κ-best, and 1/λ-best strategies that form the entire spectrum of strategies between best-so-far and iteration best and go beyond. Selecting a suitable strategy depends on the type of problem, parameters, heuristic information, and conditions in which the ACO is used. In this research, we use two representative combinatorial NP-hard problems, the symmetric traveling salesman problem (TSP) and the asymmetric traveling salesman problem (ATSP), for which very effective heuristic information is widely known, to empirically analyze the influence of strategies on the algorithmic performance. The experiments are carried out on 45 TSP and 47 ATSP instances by using the MAX-MIN ant system variant of ACO with and without local optimizations, with each problem instance repeated 101 times for 24 different pheromone reinforcement strategies. The results show that, by using adjustable pheromone reinforcement strategies, the MMAS outperformed in a large majority of cases the MMAS with classical strategies.

A Constant-factor Approximation Algorithm for the Asymmetric Traveling Salesman Problem

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem (ATSP). Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. The main idea of our approach is a reduction to Subtour Partition Cover, an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. We first show that any algorithm for Subtour Partition Cover can be turned into an algorithm for ATSP while only losing a small constant factor in the performance guarantee. Next, we present a reduction from general ATSP instances to structured instances, on which we then solve Subtour Partition Cover, yielding our constant-factor approximation algorithm for ATSP.

Improved extremal optimization for the asymmetric traveling salesman problem

This paper presents an improved extremal optimization (IEO) algorithm for solving the asymmetric traveling salesman problem (ATSP). At each update step, the IEO algorithm proceeds through two main steps: extremal dynamics and cooperative optimization. As an improvement of extremal optimization (EO), the IEO provides a general combinatorial optimization framework by emphasizing the step of cooperative optimization. In the paper, an effective cooperative optimization strategy with combination of greedy search and random walk is designed in terms of the microscopic characteristics of the ATSP solutions. Simulation results on a set of benchmark ATSP instances show that the proposed IEO algorithm provides satisfactory performance on computational effectiveness and efficiency.

Lower tolerance-based Branch and Bound algorithms for the ATSP

In this paper, we develop a new tolerance-based Branch and Bound algorithm for solving NP-hard problems. In particular, we consider the asymmetric traveling salesman problem (ATSP), an NP-hard problem with large practical relevance. The main algorithmic contribution is our lower bounding strategy that uses the expected costs of including arcs in the solution to the assignment problem relaxation of the ATSP, the so-called lower tolerance values. The computation of the lower bound requires the calculation of a large set of lower tolerances. We apply and adapt a finding from [23] that makes it possible to compute all lower tolerance values efficiently. Computational results show that our Branch and Bound algorithm exhibits very good performance in comparison with state-of-the-art algorithms, in particular for difficult clustered ATSP instances.

A hybrid approach combining an improved genetic algorithm and optimization strategies for the asymmetric traveling salesman problem

The asymmetric traveling salesman problem (ATSP) appears in various applications. Although there are several heuristic approaches to its solution, the problem is still a difficult combinatorial optimization problem. This work proposes a novel hybrid approach specialized for the ATSP. The proposed method incorporates an improved genetic algorithm (IGA) and some optimization strategies that contribute to its effectiveness. In the IGA, both the crossover operation and the mutation operation are improved by selecting the optimum from a set of solutions. Three strategies: immigration, local optimization and global optimization are established based on several empirical optimization strategies to improve the evolution of the IGA. Computational experiments are conducted on 16 ATSP instances available in the TSPLIB (traveling salesman problem library). The comparative study shows that our proposed approach outperforms several other published algorithms.

Neurodynamic programming: a case study of the traveling salesman problem

The paper focuses on the study of solving the large-scale traveling salesman problem (TSP) based on neurodynamic programming. From this perspective, two methods, temporal difference learning and approximate Sarsa, are presented in detail. In essence, both of them try to learn an appropriate evaluation function on the basis of a finite amount of experience. To evaluate their performances, some computational experiments on both the Euclidean and asymmetric TSP instances are conducted. In contrast with the large size of the state space, only a few training sets have been used to obtain the initial results. Hence, the results are acceptable and encouraging in comparisons with some classical algorithms, and further study of this kind of methods, as well as applications in combinatorial optimization problems, is worth investigating.

Evaluation of The Contract Or-Patch Heuristic Eor The Asymmetric Tsp

In this paper we further investigate tour construction algorithms for the Asymmetric Traveling Salesman Problem (ATSP). In [14] we introduced a new algorithm, called Contract-or-Patch (COP). We have tested the algorithm together with other well-known and new heuristics on a variety of families of ATSP instances. In our study, COP has demonstrated good performance, clearly outperforming all other algorithms on robustness. It has either produced the shortest tours or came close to the leader on eaeb of the seven families tested,while each of the remaining algorithms failed on at least two families of instances.In this paper we introduce three new variants of Che COP algorithm, and perform an extensive computational study of the original as well as new versions of the algorithm on a variety of ATSP instances. We also study the influence of the threshold parameter on the quality of tours produced by COP. We conclude the study by recommending one of the new versions of COP as a replacement for the original algorithm. The modified algorithm produces higher-quality tours iban the original version, and has a nice property of being a mucb simpler algorithm. We also recommend a good universal choice of the parameter value.

A note on the approximation of the asymmetric traveling salesman problem

We show that some asymmetric traveling salesman problem (ATSP) instances are approximable within bounds equal to 3 and 9/5, when they satisfy sufficient conditions more restrictive than the triangle inequality, very simple to test and nicely structured: they only depend on a measure of satisfaction of the triangle inequality and a measure of the graph asymmetry. We discuss the applicability of such conditions and we present two preprocessing linear programs to reformulate ATSP instances into equivalent ones achieving data-dependent bounds by the same approximation algorithms.