Discrete Spider Monkey Optimization for Travelling Salesman Problem

Various neural network models have been proposed to tackle combinatorial optimization problems such as the travelling salesman problem (TSP). Existing learning-based TSP methods adopt a simple setting that the training and testing data are independent and identically distributed. However, the existing literature fails to solve TSP instances when training and testing data have different distributions. Concretely, we find that different training and testing distribution will result in more difficult TSP instances, i.e., the solution obtained by the model has a large gap from the optimal solution. To tackle this problem, in this work, we study learning-based TSP methods when training and testing data have different distributions using adaptive-hardness, i.e., how difficult a TSP instance can be for a solver. This problem is challenging because it is non-trivial to (1) define hardness measurement quantitatively; (2) efficiently and continuously generate sufficiently hard TSP instances upon model training; (3) fully utilize instances with different levels of hardness to learn a more powerful TSP solver. To solve these challenges, we first propose a principled hardness measurement to quantify the hardness of TSP instances. Then, we propose a hardness-adaptive generator to generate instances with different hardness. We further propose a curriculum learner fully utilizing these instances to train the TSP solver. Experiments show that our hardness-adaptive generator can generate instances ten times harder than the existing methods, and our proposed method achieves significant improvement over state-of-the-art models in terms of the optimality gap. The codes are publicly available.

Traveling salesman problem (TSP) means that a travelling salesman needs to promote products in n cities (including the city where he lives). After visiting each city (each city can be visited once), he returns to the departure city. Let’s suppose that there is one road to connect each two cities. What is the best route to follow in order to minimize the distance of the journey? TSP has been proven to be a NP-hard problem, i.e. failure of finding a polynomial time algorithm to get a optimal solution. TSP is easy to interpret, yet hard to solve. This problem has aroused many scholars’ interests since it was put forward in 1932. However, until now, no effective solution has been found. Though TSP only represents a problem of the shortest ring road, in actual life, many physical problems are found to be the TSP. Example 1, postal route. Postal route problem is a TSP. Suppose that a mail car needs to collect mails in n places. Under such circumstances, you can show the route through a drawing containing n+1 crunodes. One crunode means a post office which this mail car departures from and returns to. The remaining n crunodes mean the crunodes at which the mails need to be collected. The route that the mail car passes through is a travelling route. We hope to find a travelling route with the shortest length. Example 2, mechanical arm. When a mechanical arm is used to fasten the nuts for the ready-to-assembling parts on the assembly line, this mechanical arm will move from the initial position (position where the first nut needs to be fastened) to each nut in proper order and then return to the initial position. The route which the mechanical arm follows is a travelling route in the drawing which contains crunodes as nuts; the most economical travelling route will enable the mechanical arm to finish its work within the shortest time. Example 3, integrated circuit. In the course of manufacturing the integrated circuit, we often need to insert thousands of electrical elements. It will consume certain energy when moving from one electrical element to the other during manufacturing. How can we do to arrange the manufacturing order to minimum the energy consumption? This is obviously a solution for TSP. Except for the above examples, problems like route distribution of transportation network, choice of tourist route, laying of pipelines needed for city planning and engineering construction are interlinked with the problems of finding the shortest route. So, it is of significance to make a study on the problem of the shortest route. This renders us a use value. As finding a solution for TSP plays an important role in the real life, since the TSP appeared, it has attracted many scholars to make a study on it.

The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, for example, algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [de Klerk E, Sotirov R (2012) Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry. Math. Programming 133(1):75–91.] present a semidefinite program (SDP) based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program but generally dominates it on small instances. We provide a family of simplicial TSP instances that shows that the integrality gap of this SDP is unbounded. Second, we show that these simplicial TSP instances imply the unbounded integrality gap of every SDP relaxation of the TSP mentioned in the survey on SDP relaxations of the TSP in section 2 of Sotirov [Sotirov R (2012) SDP relaxations for some combinatorial optimization problems. Anjos MF, Lasserre JB, eds., Handbook on Semidefinite, Conic and Polynomial Optimization (Springer, New York), 795–819.]. In contrast, the subtour linear program performs perfectly on simplicial instances. The simplicial instances thus form a natural litmus test for future SDP relaxations of the TSP.

Many real-world problems, like microchip design, can be modeled by means of the well-known traveling salesman problem (TSP). Many instances of this problem can be found in the literature. Although several optimization algorithms have been applied to TSP instances, the selection of the more promising algorithm is, in practice, a difficult decision. In this paper, a new meta-learning-based approach is investigated for the selection of optimization algorithms for TSP instances. Essentially, a learning model is trained with TSP instances for which the performance of a set of optimization algorithms is known a priori. Then, the learned model is used to predict the best algorithm for a new TSP instance. Each instance is described by meta-features that capture characteristics of the TSP that affect the performance of the optimization algorithms. Given that the best solution for a given TSP instance can be obtained by several algorithms, the meta-learning problem is considered here to be a multi-label classification problem. Several experiments illustrate the performance of the proposed approach, with promising results.

Traveling salesman problem (TSP) is a well-known NP-hard combinatorial optimization problem. It has been solved by a number of exact and approximate algorithms and serves as a testbed for new heuristic and metaheuristic optimization algorithms. However, it is often not easy to evaluate the hardness (complexity) of a TSP instance. Simple measures such as the number of cities or the minimum (maximum) route length do not capture the internal structure of a TSP instance sufficiently. In this work, we propose a new method for the assessment of TSP instance complexity based on clustering. The new approach is evaluated on a set of randomized TSP instances with different structure and its relation to the performance of a selected metaheuristic TSP solver is studied.

In the aftermath of a large disaster, the routing of vehicles carrying critical supplies can greatly impact the arrival times to those in need. Because it is critical that the deliveries are both fast and fair to those being served, it is not clear that the classic cost-minimizing routing problems properly reflect the relevant priorities in disaster relief. In this paper, we take the first steps toward developing new methodologies for these problems. We focus specifically on two alternative objective functions for the traveling salesman problem (TSP) and the vehicle routing problem (VRP): one that minimizes the maximum arrival time (minmax) and one that minimizes the average arrival time (minavg). To demonstrate the potential impact of using these new objective functions, we bound the worst-case performance of optimal TSP solutions with respect to these new variants and extend these bounds to include multiple vehicles and vehicle capacity. Similarly, we examine the potential increase in routing costs that results from using these alternate objectives. We present solution approaches for these two variants of the TSP and VRP, which are based on well-known insertion and local search techniques. These are used in a series of computational experiments to help identify the types of instances in which TSP and VRP solutions can be significantly different from optimal minmax and minavg solutions.

The goal of Traveling Salesman Problem (TSP) is to find the shortest circular tour visiting every city exactly once. TSP has many real world applications and a number of methods have been investigated to solve TSP. Recently, nature inspired algorithms are also attracted to solve it. Here we studied Group Search Optimizer(GSO), the recently proposed nature inspired algorithm, to solve TSP. GSO is a population based optimization technique on the metaphor of producer-scrounger based social behavior of animals where producer searches for finding foods and scrounger searches for joining opportunities. GSO has found as an efficient method for solving function optimization problems for which it modeled. In this study we employ the concept of Swap Operator (SO) and Swap Sequence (SS) to modify GSO for TSP. The modified GSO (mGSO) was tested on a number of benchmark TSPs and results compared with some existing approaches. mGSO has shown best results (best tour cost) for some problems and competitive performance in other cases.

The Traveling Salesman Problem (TSP) is a combinatorial optimization problem which has a plethora of real-world applications in various domains. Since large scale TSP problems are difficult to be solved by deterministic algorithms in a reasonable amount of time, heuristic and meta-heuristic algorithms such as swarm intelligence algorithms are usually used for solving TSP. These algorithms provide near-optimal solutions in a feasible amount of time. The Dragonfly Algorithm (DA) is a recent swarm intelligence algorithm and it has shown to have a higher performance as compared to other swarm intelligence algorithms in various applications. Since the original DA algorithm is proposed for solving continuous optimization problems, it cannot be used for solving TSP and although a binary version of the algorithm, called BDA, is also proposed, it is not suitable for solving TSP. Hence, in this paper, a variant of the DA algorithm is proposed. DA is adapted for solving TSP by adapting its equations and making use of the method of swap sequences to update the position of the artificial dragonflies in the search space. The algorithm is employed to solve a TSP problem which consists of four cities and the results show that the proposed algorithm is able to provide the optimal TSP path.

In this paper, triangular intuitionistic fuzzy travelling salesman problem has been considered to identify the shortest distance between the Cities. The Dynamic Programming method is applied to obtain the solution of the intuitionistic fuzzy travelling salesman problem without converting them into a crisp Travelling Salesman Problem. For a better understanding, numerical examples are also provided.

We propose an end-to-end learning framework based on hierarchical reinforcement learning, called H-TSP, for addressing the large-scale Traveling Salesman Problem (TSP). The proposed H-TSP constructs a solution of a TSP instance starting from the scratch relying on two components: the upper-level policy chooses a small subset of nodes (up to 200 in our experiment) from all nodes that are to be traversed, while the lower-level policy takes the chosen nodes as input and outputs a tour connecting them to the existing partial route (initially only containing the depot). After jointly training the upper-level and lower-level policies, our approach can directly generate solutions for the given TSP instances without relying on any time-consuming search procedures. To demonstrate effectiveness of the proposed approach, we have conducted extensive experiments on randomly generated TSP instances with different numbers of nodes. We show that H-TSP can achieve comparable results (gap 3.42% vs. 7.32%) as SOTA search-based approaches, and more importantly, we reduce the time consumption up to two orders of magnitude (3.32s vs. 395.85s). To the best of our knowledge, H-TSP is the first end-to-end deep reinforcement learning approach that can scale to TSP instances of up to 10000 nodes. Although there are still gaps to SOTA results with respect to solution quality, we believe that H-TSP will be useful for practical applications, particularly those that are time-sensitive e.g., on-call routing and ride hailing service.

The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard combinatorial optimization problems (COPs). Many social production problems can be effectively represented as instances of TSPs. However, solving large-scale TSPs remains a significant challenge for conventional Von Neumann computers. Many studies have proposed annealing processors to address large-scale COPs, but most of them focus on unconstrained problems, such as the Maxcut problem. In this paper, a scalable decoupled-clustering annealng processor (DCAP) for efficiently handling large-scale TSPs is presented. A decoupled hierarchical clustering algorithm is proposed for higher convergence speed and improved scalability. Several techniques have been developed in hardware to minimize area overhead and processing time, including a modified spin connection topology for the Ising model, an area-efficient random threshold generator, a one-step spin update scheme and a dynamic prediction method. The DCAP prototype is implemented on FPGA with an operating frequency of 125MHz. We tested our design on various TSP instances from the TSPLIB. Results show that our design outperforms the CPU- and GPU-based Neuro-Ising scheme by achieving maximum speedups of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$780\times $ </tex-math></inline-formula> and a 42% improvement in accuracy. With multi-chip interconnection, DCAP is able to handle problems of scale up to 85900 cities.

In this paper, we investigate some modified local search (LS) heuristics for the solution of symmetric traveling salesman problems (TSPs). These modifications are mainly due to the use of extended neighborhood structures. In addition, we are concerned with several new sets of the moves (transitions of solutions) based on the extended configurations of edge exchanges. We are also examining the performance of these extensions being used in an iterated local search (ILS) paradigm. The results from the experiments with the benchmark TSP instances from the TSP library (TSPLIB) demonstrate that the introduced improvements enable to seek solutions of higher quality without substantially increasing computational complexity.DOI: http://dx.doi.org/10.5755/j01.itc.42.3.1301

It is prone to get stuck in a local minimum when solving the Traveling Salesman Problem (TSP) by the traditional Hopfield neural network (HNN) and hard to converge to an efficient solution, resulting from the defect of the penalty method used by the HNN. In order to mend this defect, an accelerated augmented Lagrangian Hopfield neural network (AALHNN) algorithm was proposed in this paper. This algorithm gets out of the dilemma of penalty method by Lagrangian multiplier method, ensuring that the solution to the TSP is undoubtedly efficient. The second order factor added in the algorithm stabilizes the neural network dynamic model of the problem, thus improving the efficiency of solution. In this paper, when solving the TSP by AALHNN, some changes were made to the TSP models of Hopfield and Tank. Say, constraints of TSP are multiplied by Lagrange multipliers and augmented Lagrange multipliers respectively, The augmented Lagrange function composed of path length function can ensure robust convergence and escape from the local minimum trap. The Lagrange multipliers are updated by using nesterov acceleration technique. In addition, it was theoretically proved that the extremum obtained by this improved algorithm is the optimal solution of the initial problem and the approximate optimal solution of the TSP was successfully obtained several times in the simulation experiment. Compared with the traditional HNN, this method can ensure that it is effective for TSP solution and the solution to the TSP obtained is better.

Objectives: A well-known NP-complete problem is the travelling salesman problem (TSP). It has numerous engineering and scientific applications. In this article, we have proposed a multi-conveyance TSP where different conveyances are present to travel from one city to another city. This is an extension to classical TSP. In this TSP, the salesman visits all the cities only once during his/her tour, using different conveyances to travel from one city to another. The cost of travelling between cities using various modes of conveyance varies. The objective of this research is to find the minimum cost tour using an ant colony optimization (ACO) based approach by satisfying the constraints of the proposed multi-conveyance TSP. Method: The considered TSP has been solved using a novel ACO technique. The proposed ACO is adopted with the Roulette-wheel selection and “tuning solution” techniques. We have used a few benchmark datasets from TSPLIB to check the effectiveness of the proposed algorithm. The experimental findings for a few benchmarks TSP instances show that almost always, the proposed ACO is able to find a better result. Then, we used some redefined and randomly generated datasets for experiments. The experimental outcomes for various input datasets are also very encouraging. Findings: The goal of the proposed TSP is to find a complete tour with a minimum cost without exceeding the total travel cost and total travel time. Thus, there are two novel constraints in the classical TSP. Novelty: A unique aspect of the proposed research is the use of several conveyance facilities and a fixed total travel time and cost. This is new, as these three factors are integrated into a single TSP model. Keywords: Travelling Salesman Problem; Travel Cost; Travel Time; MultiConveyance TSP; Ant Colony Optimization

In this paper, we present some new results obtained for the traveling salesman problem (TSP) by using the iterated tabu search (ITS) meta-heuristic. ITS is a promising extension to the ordinary tabu search scheme. It seeks near-optimal solutions by combining intensification (standard tabu search) and diversification (perturbation of solutions) in a proper way. For the TSP, the main effect is achieved due to decomposition of the solution neighbourhood structure and considerably speeding-up the tabu search process, which is used, namely, in the role of intensification. This fast-iterated tabu search (FITS) technique resulted in quite encouraging solutions for the TSP instances from the TSP instance library TSPLIB. FITS obviously outperformed the other heuristic algorithms used in the experimentation, especially, on the smaller TSP instances.