A Constructive Self-Organizing Network Applied to a Discrete Optimization Problem

Results on human performance on the Traveling Salesman Problem (TSP) from different laboratories show high consistency. However, one exception is in the area of individual differences. While one research group has consistently failed to find systematic individual differences across instances of TSPs (Chronicle, MacGregor and Ormerod), another group (Vickers, Lee and associates) has found individual differences both within TSP performance and between TSP performance and other cognitive tasks. Among possible reasons for the conflicting results are differences in procedure and differences in the problem instances used. To try to resolve the discrepancy, we collected data on TSP performance by combining the procedure used by one group with problem instances used by the other. The comparison involved nine 30-node and nine 40-node TSP problems previously used by the Vickers group, using computer presentation. Here, we had the same problems completed by 112 participants using a paper-and-pencil mode of presentation. We examined the results in the form of distributions of correlations across individuals for each pair of problems of the same size. The distributions for the computer and paper forms of presentation were very similar, and centered between correlations of 0.20 and 0.30. The results indicated the presence of individual differences at a level that fell between those previously reported by the two laboratories. The pattern of results indicated that previous discrepancies did not arise because of differences in procedure. Instead, individual differences appeared to become more prevalent as the difficulty of problems increased. The results are consistent with an explanation that performance on simpler instances is dominated by lower-level processes, but that as instance difficulty increases, higher-level functions become increasingly involved.

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.

<p indent="0mm">As a high-performance general-purpose form of parallel solvers, parallel algorithm portfolios (PAPs) have recently shown great performance for solving decision, counting, continuous, and discrete optimization problems. However, the manual construction of PAPs is laborious work, which typically requires substantial domain knowledge and human effort. To address this issue, this paper proposes AutoPAP, an evolutionary intelligence-based method for PAP construction. Overall, AutoPAP follows the (<italic>n</italic>+1) evolutionary framework, i.e., in each generation, <italic>n</italic> candidate algorithms are generated, and the best one among them is retained in the PAP. Considering that the algorithm configuration space is typically very large, this study designs a highly effective mutation operator to improve the practical performance of AutoPAP and further theoretically proves that AutoPAP can achieve (1−1/<italic>e</italic>) approximation. Finally, to validate the effectiveness of AutoPAP, this study uses it to build a PAP, namely, TSP_PAP, for traveling salesman problems (TSPs). The test results show that TSP_PAP significantly outperforms state-of-the-art TSP solvers EAX and LKH in terms of efficiency (runtime) and effectiveness (solution quality). On 128 TSP instances of sizes ranging from 1000 to 30000, compared with LKH and EAX, TSP_PAP can reduce the average runtime by at least 45.71% and lower the average deviation ratio by at least 87.50%, indicating the huge potential of AutoPAP in automatic algorithm design and evolution.

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.

Genetic algorithms (GAs) are search algorithms based on population genetics and natural selection concepts. Maintaining population variety in GAs is critical for ensuring global exploration and mitigating the risks of premature convergence. Rapid convergence to local optima is one such challenge in the application of genetic algorithms. To address this issue, we provide Cave-Surface GA (CSGA), an alternative method based on the Dual Population Genetic Algorithm and inspired by the genetic variety observed in Mexican cavefish. Through inter-population crossbreeding, CSGA increases diversity via a secondary population (Cave population) and facilitates the exchange of information between populations, effectively counteracting premature convergence. Several experiments are carried out utilizing benchmark instances of the Traveling Salesman Problem (TSP) obtained from TSPLIB, a well-known TSP problem library. Our experimental results over many TSP instances show that CSGA outperforms both classic GAs and other GAs that use diversity preservation techniques, such as Multipopulation GA (MPGA). CSGA has the potential to give promising solutions to challenging optimization issues like TSP.

This paper is concerned with the discrete combinatorial optimization problem. A novel set-based differential evolution (SBDE) algorithm for traveling salesman problem (TSP) is proposed and evaluated. SBDE is based on a set-based representation scheme that enables SBDE to characterize the discrete search space of TSP. The candidate solution, vector and the arithmetical operator in SBDE are redefined. Numerical experiments are carried out using the TSPLIB of test TSP problems. The obtained results show that the proposed approach is effective in quality for solving TSP.

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.

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.

Solving the traveling salesman problem based on an adaptive simulated annealing algorithm with greedy search

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.

We show how, by a constructive process, we can generate arbitrarily large instances of the Traveling Salesman Problem (TSP) using standard fractals such as those of Peano, Koch, or Sierpinski. We show that optimal solutions for these TSPs can be known a priori, and thus, they provide us with new nontrivial TSP instances offering the possibility of testing heuristics well beyond the scope of testbed instances which have been solved by exact numerical methods. Furthermore, instances may be constructed with different features, for example, with different fractal dimensions. To four of these fractal TSPs we apply three standard constructive heuristics, namely Multiple Fragment, Nearest Neighbor, and Farthest Insertion from Convex-Hull, which have efficient general-purpose implementations. The ability of different algorithms to solve these different fractal TSPs gives us significant insight into the nature of TSP heuristics in a way which is complementary to approaches such as worst-case or average-case analysis.

The Traveling Salesman Problem (TSP) is an NP-hard graph search problem. Despite having numerous modifications of the original abstract problem, Time Dependent Traveling Salesman Problem (TD TSP) was one of the most realistic extensions under real traffic conditions. In TD TSP the edges between nodes are assigned higher costs (weights), if they were traveled during the rush hour periods, or crossed the traffic jam regions, such as the city center(s). In this paper we introduce an even more real-life motivated approach, the Intuitionistic Fuzzy Time Dependent Traveling Salesman Problem (IFTD TSP), which is a further extension of the TSP, and also of the classic TD TSP, with the additional notion of using intuitionistic fuzzy sets for the definition of uncertain costs, time, and space of the rush hour—traffic jam region affecting graph sections. In IFTD TSP we use fuzzy memberships and non-memberships sets for estimating the vague costs between nodes in order to quantify the behavior of traffic jam regions, and the rush hour periods. Since intuitionistic fuzzy sets are generalizations of classic fuzzy sets, our approach may be considered an extension and substitution of the original abstract TD TSP problem, even, of the (classic) Fuzzy TD TSP. Lastly, DBMEA (Discrete Bacterial Memetic Evolutionary Algorithm) was applied on the IFTD TSP model, the results of the simulation runs based on some extensions of the benchmarks generated from the original TD TSP data set showed quite good and promising preliminary results.

Discrete Spider Monkey Optimization for Travelling Salesman Problem

The travelling salesman problem (TSP) is a popular NP-hard combinatorial optimization problem that requires finding the optimal way for a salesman to travel through different cities once and return to the initial city. The existing methods of solving TSPs on quantum systems are either gate-based or binary variable-based encoding. Both approaches are resource-expensive in terms of the number of qubits, while performing worse compared to existing classical algorithms, even for small-sized problems.&amp;#xD;A novel encoding scheme is needed to map the TSP problem onto a quantum system, which is addressed in this work.&amp;#xD;We introduce a distinct geometric approach to encode the TSP on a single qubit and present a quantum-inspired algorithm to solve the problem by invoking the principle of \textit{quantum superposition}.&amp;#xD;The cities are represented as quantum states on the Bloch sphere, while the preparation of superposition states allows us to traverse multiple paths at once. The underlying framework of our algorithm is a quantum-inspired version of the classical Brachistochrone approach.&amp;#xD;Optimal control methods are employed to create a selective superposition of the quantum states to find the shortest route of a given TSP.&amp;#xD;The numerical simulations solve a sample of four to nine cities for which exact solutions are obtained. The algorithm can be implemented on any quantum platform capable of efficiently rotating a qubit and allowing state tomography measurements. For the TSP problem sizes considered in this work, our algorithm is more resource-efficient and accurate than existing quantum algorithms, with the potential for scalability.

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.