Combinatorial Optimization Research Articles

Source code obfuscation with genetic algorithms using LLVM code optimizations

Abstract With the advent of the cloud computing model allowing a shared access to massive computing facilities, a surging demand emerges for the protection of the intellectual property tied to the programs executed on these uncontrolled systems. If novel paradigm as confidential computing aims at protecting the data manipulated during the execution, obfuscating techniques (in particular at the source code level) remain a popular solution to conceal the purpose of a program or its logic without altering its functionality, thus preventing reverse-engineering on the program even with the help of computing resources. The many advantages of code obfuscation, together with its low cost, makes it a popular technique. This paper proposes a novel methodology for source code obfuscation that can be used together with other traditional obfuscation techniques, making the code more robust against reverse engineering attacks. Three program complexity metrics are used to define three different single-objective combinatorial optimization versions of the problem, which are solved and analysed. Additionally, three multi-objective problems are defined, those considering each of the selected metrics together with the program execution time, in order to avoid strong obfuscations penalizing the performance. The goal of the defined problems is to find sequences of LLVM optimizations that lead to highly obfuscated versions of the original code. These transformations are applied to the back-end pseudo-assembly code (i.e., LLVM Intermediate Representation), thus avoiding any further optimizations by the compiler. Classical genetic algorithms (GAs) are used to solve the studied problems, namely a basic cellular GA for the single-objective problems and the popular NSGA-II for the multi-objective ones. The promising results show the potential of the proposed technique.

Estimation of distribution algorithms with solution subset selection for the next release problem

Abstract The Next Release Problem (NRP) is a combinatorial optimization problem that aims to find a subset of software requirements to be delivered in the next software release, which maximize the satisfaction of a list of clients and minimize the effort required by developers to implement them. Previous studies have applied various metaheuristics, mostly genetic algorithms. Estimation of Distribution Algorithms (EDA), based on probabilistic modelling, have been proved to obtain good results in problems where genetic algorithms struggle. In this paper we propose to adapt three EDAs to tackle the multi-objective NRP in a fast and effective way. Results show that EDAs can be applicable to solve the NRP with rather good quality of solutions. Furthermore, we prove that their execution time can be significantly reduced using a per-iteration solution subset selection method while maintaining the overall quality of the solutions obtained, and they perform the best when limiting the search time as in an interactive tool that requires fast responsiveness. The experimental framework, code and datasets have been made public in a code repository.

Optimal selection of manufacturing enterprise financialization based on simulated annealing investment portfolio theory

The global market competition is becoming increasingly fierce, and manufacturing enterprises need to invest and expand. However, the traditional financial optimization of manufacturing enterprises has faced problems such as low efficiency and inaccurate search for the optimal solution, which has made manufacturing enterprises likely to face financial risks. The investment portfolio can enable enterprises to obtain the maximum profit on a certain risk level, or reduce their investment risk on a certain return level as far as possible. If the combinatorial optimization is realized, it can be applied to the optimal selection of manufacturing enterprises’ financialization. This article analyzed the respective characteristics of Genetic Algorithm (GA) and Simulated Annealing (SA) algorithms, and analyzed the combination of GA and SA algorithms to solve the optimal investment portfolio through GA-SA algorithm, thereby helping manufacturing enterprises to make the optimal choice for financialization. The experimental results of this article indicated that the GA-SA algorithm solved the problem of GA algorithm easily falling into local optima, SA algorithm’s initial temperature and generation mechanism, and improved the efficiency of finding the optimal solution. Meanwhile, the experimental results showed that the average optimal solutions of Genetic Algorithm, Simulated Annealing algorithm, and GA-SA algorithms for 36 stock portfolios in Enterprise 1 were 69, 69, and 107, respectively. The average optimal solutions of the three algorithms for 36 stock portfolios in Enterprise 2 were 73, 90, and 112, respectively. This proves that the number of optimal solutions searched by GA-SA algorithm is higher than that of GA and SA algorithm, and also proves that it is effective to use GA-SA algorithm to optimize investment portfolio and help manufacturing enterprises to make optimal financial choices.

Research on Multi-Objective Flexible Job Shop Scheduling Problem with Setup and Handling Based on an Improved Shuffled Frog Leaping Algorithm

Flexible job shop scheduling problem (FJSP), widely prevalent in many intelligent manufacturing industries, is one of the most classic problems of production scheduling and combinatorial optimization. In actual manufacturing enterprises, the setup of machines and the handling of jobs have an important impact on the scheduling plan. Furthermore, there is a trend for a cluster of machines with similar functionalities to form a work center. Considering the above constraints, a new order-driven multi-equipment work center FJSP model with setup and handling including multiple objectives encompassing the minimization of the makespan, the number of machine shutdowns, and the number of handling batches is established. An improved shuffled frog leading algorithm is designed to solve it through the optimization of the initial solution population, the improvement of evolutionary operations, and the incorporation of Pareto sorting. The algorithm also combines the speed calculation method in the gravity search algorithm to enhance the stability of the solution search. Some standard FJSP data benchmarks have been selected to evaluate the effectiveness of the algorithm, and the experimental results confirm the satisfactory performance of the proposed algorithm. Finally, a problem example is designed to demonstrate the algorithm’s capability to generate an excellent scheduling plan.

Quantum-enhanced hybrid feature engineering in thoracic CT image analysis for state-of-the-art nodule classification: an advanced lung cancer assessment

The intricate nature of lung cancer treatment poses considerable challenges upon diagnosis. Early detection plays a pivotal role in mitigating its escalating global mortality rates. Consequently, there are pressing demands for robust and dependable early detection and diagnostic systems. However, the technological limitations and complexity of the disease make it challenging to implement an efficient lung cancer screening system. AI-based CT image analysis techniques are showing significant contributions to the development of computer-assisted detection (CAD) systems for lung cancer screening. Various existing research groups are working on implementing CT image analysis systems for assessing and classifying lung cancer. However, the complexity of different structures inside the CT image is high and comprehension of significant information inherited by them is more complex even after applying advanced feature extraction and feature selection techniques. Traditional and classical feature selection techniques may struggle to capture complex interdependencies between features. They may get stuck in local optima and sometimes require additional exploration strategies. Traditional techniques may also struggle with combinatorial optimization problems when applied to a prominent feature space. This paper proposed a methodology to overcome the existing challenges by applying feature extraction using Vision Transformer (FexViT) and Feature selection using the Quantum Computing based Quadratic unconstrained binary optimization (QC-FSelQUBO) technique. This algorithm shows better performance when compared with other existing techniques. The proposed methodology showed better performance as compared to other existing techniques when evaluated by applying necessary output measures, such as accuracy, Area under roc (receiver operating characteristics) curve, precision, sensitivity, and specificity, obtained as 94.28%, 99.10%, 96.17%, 90.16% and 97.46%. The further advancement of CAD systems is essential to meet the demand for more reliable detection and diagnosis of cancer, which can be addressed by leading the proposed quantum computation and growing AI-based technology ahead.

Time-reliability optimization for the stochastic traveling salesman problem

This paper presents a novel approach to addressing the Stochastic Traveling Salesman Problem (STSP), a classical problem in combinatorial optimization, by integrating travel time and reliability factors into the decision-making process. Traditional TSP models primarily focus on minimizing the total travel distance or cost without considering the reliability of each route. In real-world situations, especially in logistics and network design, it's just as important to have reliable routes. A reliable route means there's a good chance it will be completed successfully and on time. Our research extends the conventional STSP framework by incorporating a reliability metric for each route, alongside the standard travel time metric. A tri-objective optimization model is proposed to minimize the mean and standard deviation of travel time and maximize route reliability simultaneously. A new algorithm called Permutation Binary-Addition-Tree (BAT) is proposed to solve the problem more efficiently when there is uncertainty. Our approach marks a significant step towards more realistic and practical solutions for route optimization problems in dynamic and uncertain environments. We also present a complexity analysis of our model against traditional cost-only TSP solutions, demonstrating the efficacy of considering reliability in route planning.

Pixelation of time matrices for solving permutation flowshop scheduling problem

Over the last decades, scheduling theory has been put into practice so as to tackle production process as a combinatorial optimisation problem and specifically as a PFSP (Permutation Flowshop Scheduling Problem). Often, NP-completeness of PFSP leads numerous research towards suggesting heuristic algorithms. The objective is to propose an optimal or near-to-optimal order of n jobs processing on m machines with a minimum completion time of all jobs. In this paper, a two-phase heuristic algorithm will be presented, named IFRS (Improved FRS). FRS, within the first phase, is going to be used to find a superior order of jobs out of Taillard’s instances; then the order of jobs will be improved again in phase 2. While running IFRS, a completely new idea, named Pixelation of time matrices, will be used for the very first time and provide a final pattern, which can be used in any heuristic algorithm with makespan criterion. After using the hard benchmark instances of Taillard, the superiority of IFRS over 20 algorithms is going to be shown by tables and statistical graphics. Due to its low makespan values, IFRS benefits flow production, as does NEH.

Explanations for combinatorial optimization problems

We introduce a representation for generating explanations for the outcomes of combinatorial optimization algorithms. The two key ideas are (A) to maintain fine-grained representations of the values manipulated by these algorithms and (B) to derive explanations from these representations through merge, filter, and aggregation operations. An explanation in our model presents essentially a high-level comparison of the solution to a problem with a hypothesized alternative, illuminating why the solution is better than the alternative. Our value representation results in explanations smaller than other dynamic program representations, such as traces. Based on a measure for the conciseness of explanations we demonstrate through a number of experiments that the explanations produced by our approach are small and scale well with problem size across a number of different applications.

Quantum-Informed Recursive Optimization Algorithms

We propose and implement a family of quantum-informed recursive optimization (QIRO) algorithms for combinatorial optimization problems. Our approach leverages quantum resources to obtain information that is used in problem-specific classical reduction steps that recursively simplify the problem. These reduction steps address the limitations of the quantum component (e.g., locality) and ensure solution feasibility in constrained optimization problems. Additionally, we use backtracking techniques to further improve the performance of the algorithm without increasing the requirements on the quantum hardware. We showcase the capabilities of our approach by informing QIRO with correlations from classical simulations of shallow circuits of the quantum approximate optimization algorithm, solving instances of maximum independent set and maximum satisfiability problems with hundreds of variables. We also demonstrate how QIRO can be deployed on a neutral atom quantum processor to find large independent sets of graphs. In summary, our scheme achieves results comparable to classical heuristics even with relatively weak quantum resources. Furthermore, enhancing the quality of these quantum resources improves the performance of the algorithms. Notably, the modular nature of QIRO offers various avenues for modifications, positioning our work as a template for a broader class of hybrid quantum-classical algorithms for combinatorial optimization. Published by the American Physical Society 2024

A deep reinforcement learning algorithm framework for solving multi-objective traveling salesman problem based on feature transformation

As a special type of multi-objective combinatorial optimization problems (MOCOPs), the multi-objective traveling salesman problem (MOTSP) plays an important role in practical fields such as transportation and robot control. However, due to the complexity of its solution space and the conflicts between different objectives, it is difficult to obtain satisfactory solutions in a short time. This paper proposes an end-to-end algorithm framework for solving MOTSP based on deep reinforcement learning (DRL). By decomposing strategies, solving MOTSP is transformed into solving multiple single-objective optimization subproblems. Through linear transformation, the features of the MOTSP are combined with the weights of the objective function. Subsequently, a modified graph pointer network (GPN) model is used to solve the decomposed subproblems. Compared with the previous DRL model, the proposed algorithm can solve all the subproblems using only one model without adding weight information as input features. Furthermore, our algorithm can output a corresponding solution for each weight, which increases the diversity of solutions. In order to verify the performance of our proposed algorithm, it is compared with four classical evolutionary algorithms and two DRL algorithms on several MOTSP instances. The comparison shows that our proposed algorithm outperforms the compared algorithms both in terms of training time and the quality of the resulting solutions.

MEMS Oscillators-Network-Based Ising Machine with Grouping Method.

Combinatorial optimization (CO) has a broad range of applications in various fields, including operations research, computer science, and artificial intelligence. However, many of these problems are classified as nondeterministic polynomial-time (NP)-complete or NP-hard problems, which are known for their computational complexity and cannot be solved in polynomial time on traditional digital computers. To address this challenge, continuous-time Ising machine solvers have been developed, utilizing different physical principles to map CO problems to ground state finding. However, most Ising machine prototypes operate at speeds comparable to digital hardware and rely on binarizing node states, resulting in increased system complexity and further limiting operating speed. To tackle these issues, a novel device-algorithm co-design method is proposed for fast sub-optimal solution finding with low hardware complexity. On the device side, a piezoelectric lithium niobate (LiNbO3) microelectromechanical system (MEMS) oscillator network-based Ising machine without second-harmonic injection locking (SHIL) is devised to solve Max-cut and graph coloring problems. The LiNbO3 oscillator operates at speeds greater than 9 GHz, making it one of the fastest oscillatory Ising machines. System-wise, an innovative grouping method is used that achieves a performance guarantee of 0.878 for Max-cut and 0.658 for graph coloring problems, which is comparable to Ising machines that utilize binarization.

Particle Swarm Optimization Based Feature Selection Using Factorial Design

Feature selection, a common and crucial problem in current scientific research, is a crucial data preprocessing technique and a combinatorial optimization task. Feature selection aims to select a subset of informative and appropriate features from the original feature dataset. Therefore, improving performance on the classification task requires processing the original data using a feature selection strategy before the learning process. Particle swarm optimization, one of the metaheuristic algorithms that prevents the growth of computing complexity, can solve the feature selection problem satisfactorily and quickly with appropriate classification accuracy since it has local optimum escape strategies. There are arbitrary trial and error approaches described separately in the literature to determine the critical binary particle swarm optimization parameters, which are the inertial weight, the transfer function, the threshold value, and the swarm size, that directly affect the performance of the binary particle swarm optimization algorithm parameters used in feature selection. Unlike these approaches, this paper enables us to obtain scientific findings by evaluating all binary particle swarm optimization parameters together with the help of a statistically based factorial design approach. The results show how well the threshold and the transfer function have statistically affected the binary particle swarm optimization algorithm performance.

Q-learning guided algorithms for bi-criteria minimization of total flow time and makespan in no-wait permutation flowshops

Combining Deep Reinforcement Learning and meta-heuristic techniques represents a new research direction for enhancing the search capabilities of meta-heuristic methods in the context of production scheduling. Q-learning is a prominent reinforcement learning in which its utilization aims to direct the selection of actions, thus preventing the necessity for a random exploration in the iterative process of the metaheuristics. In this study, we provide Q-learning guided algorithms for the Bi-Criteria No-Wait Flowshop Scheduling Problem (NWFSP). The problem is treated as a bi-criteria combinatorial optimization problem where total flow time and makespan are optimized simultaneously. Firstly, a deterministic mixed-integer linear programming (MILP) model is provided. Then, Q-learning guided algorithms are developed: Bi-Criteria Iterated Greedy Algorithm with Q-Learning (BC-IGQL). Bi-Criteria Block Insertion Heuristic Algorithm with Q-Learning (BC-BIHQL). Moreover, the performance of the proposed Q-learning guided algorithms is compared over a collection of Bi-Criteria Genetic Local Search Algorithms (BC-GLS), Bi-Criteria Iterated Greedy Algorithm (BC-IG), Bi-Criteria Iterated Greedy Algorithm with a Local Search (BC-IGALL) and Bi-Criteria Variable Block Insertion Heuristic Algorithm (BC-VBIH). The complete computational experiment, performed on 480 problem instances of Vallada et al. (2015), which is known as the VRF benchmark set, indicates that the BC-BIHQL and the BC-IGQL algorithms outperform the BC-GLS, BC-IG, BC-IGALL, and BC-VBIH algorithms in comparative performance metrics. More specifically, the proposed BC-BIHQL and BC-IGQL algorithms can yield more non-dominated bi-criteria solutions with the most substantial competitiveness than the remaining algorithms. At the same time, both are competitive with each other on the benchmark problems. Moreover, the BC-IGQL algorithm dominates almost 97% and 99% of the solutions reached by the BC-IG, BC-IGALL, and BC-VBIH algorithms in small and large datasets. Similarly, The BC-BIHQL algorithm dominates almost 98% and 99% of the solutions reached by the BC-IG, BC-IGALL, and BC-VBIH algorithms in small and large datasets, respectively. This means that, among all the features that have been compared, the Q-learning-guided algorithms demonstrate the highest level of competitiveness. The outcomes of this study encourage us to discover many more bi-criteria NWFSPs to reveal the trade-off between other conflicting objectives, such as makespan & the number of early jobs, to overcome various industries' problems.

Capacitated Clustering via Majorization-Minimization and Collaborative Neurodynamic Optimization.

This paper addresses capacitated clustering based on majorization-minimization and collaborative neurodynamic optimization (CNO). Capacitated clustering is formulated as a combinatorial optimization problem. Its objective function consists of fractional terms with intra-cluster similarities in their numerators and cluster cardinalities in their denominators as normalized cluster compactness measures. To obviate the difficulty in optimizing the objective function with factional terms, the combinatorial optimization problem is reformulated as an iteratively reweighted quadratic unconstrained binary optimization problem with a surrogate function and a penalty function in a majorization-minimization framework. A clustering algorithm is developed based on CNO for solving the reformulated problem. It employs multiple Boltzmann machines operating concurrently for local searches and a particle swarm optimization rule for repositioning neuronal states upon their local convergence. Experimental results on ten benchmark datasets are elaborated to demonstrate the superior clustering performance of the proposed approaches against seven baseline algorithms in terms of 21 internal cluster validity criteria.

Advancing feature ranking with hybrid feature ranking weighted majority model: a weighted majority voting strategy enhanced by the Harris hawks optimizer

Abstract Feature selection (FS) is vital in improving the performance of machine learning (ML) algorithms. Despite its importance, identifying the most important features remains challenging, highlighting the need for advanced optimization techniques. In this study, we propose a novel hybrid feature ranking technique called the Hybrid Feature Ranking Weighted Majority Model (HFRWM2). HFRWM2 combines ML models with the Harris Hawks Optimizer (HHO) metaheuristic. HHO is known for its versatility in addressing various optimization challenges, thanks to its ability to handle continuous, discrete, and combinatorial optimization problems. It achieves a balance between exploration and exploitation by mimicking the cooperative hunting behavior of Harris’s hawks, thus thoroughly exploring the search space and converging toward optimal solutions. Our approach operates in two phases. First, an odd number of ML models, in conjunction with HHO, generate feature encodings along with performance metrics. These encodings are then weighted based on their metrics and vertically aggregated. This process produces feature rankings, facilitating the extraction of the top-K features. The motivation behind our research is 2-fold: to enhance the precision of ML algorithms through optimized FS and to improve the overall efficiency of predictive models. To evaluate the effectiveness of HFRWM2, we conducted rigorous tests on two datasets: “Australian” and “Fertility.” Our findings demonstrate the effectiveness of HFRWM2 in navigating the search space and identifying optimal solutions. We compared HFRWM2 with 12 other feature ranking techniques and found it to outperform them. This superiority was particularly evident in the graphical comparison of the “Australian” dataset, where HFRWM2 showed significant advancements in feature ranking.

Binary matrix factorization via collaborative neurodynamic optimization

Binary matrix factorization is an important tool for dimension reduction for high-dimensional datasets with binary attributes and has been successfully applied in numerous areas. This paper presents a collaborative neurodynamic optimization approach to binary matrix factorization based on the original combinatorial optimization problem formulation and quadratic unconstrained binary optimization problem reformulations. The proposed approach employs multiple discrete Hopfield networks operating concurrently in search of local optima. In addition, a particle swarm optimization rule is used to reinitialize neuronal states iteratively to escape from local minima toward better ones. Experimental results on eight benchmark datasets are elaborated to demonstrate the superior performance of the proposed approach against six baseline algorithms in terms of factorization error. Additionally, the viability of the proposed approach is demonstrated for pattern discovery on three datasets.

Extension of CMSA with a Learning Mechanism: Application to the Far from Most String Problem

One of the problems with exact techniques for solving combinatorial optimization problems is that they tend to run into problems with growing problem instance size. Nevertheless, they might still be very usefully employed, even in the context of large problem instances, as a sub-ordinate method within so-called hybrid metaheuristics. “Construct, Merge, Solve and Adapt” (Cmsa) is a hybrid metaheuristic technique that allows the application of exact methods to large-scale problem instances through intelligent instance reduction. However, Cmsa does not make use of an explicit learning mechanism. In this work, an algorithm called LEARN_CMSA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsc {Learn}\\_\ extsc {Cmsa}$$\\end{document} is presented for the application to the far from most string problem (FFMSP), which is an NP-hard combinatorial optimization problem from the field of string consensus problems. LEARN_CMSA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsc {Learn}\\_\ extsc {Cmsa}$$\\end{document} results from hybridization between Cmsa and a population-based algorithm. By means of this hybridization, explicit learning is introduced to Cmsa. Even though the FFMSP is a well-studied problem, LEARN_CMSA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsc {Learn}\\_\ extsc {Cmsa}$$\\end{document} achieves superior performance when compared to current state-of-the-art solvers.

Restricted global optimization for QAOA

The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising variational quantum algorithm for addressing NP-hard combinatorial optimization problems. However, a significant limitation lies in optimizing its classical parameters, which is in itself an NP-hard problem. To circumvent this obstacle, initialization heuristics, enhanced problem encodings and beneficial problem scalings have been proposed. While such strategies further improve QAOA’s performance, their remaining problem is the sole utilization of local optimizers. We show that local optimization methods are inherently inadequate within the complex cost landscape of QAOA. Instead, global optimization techniques greatly improve QAOA’s performance across diverse problem instances. While global optimization generally requires high numbers of function evaluations, we demonstrate how restricted global optimizers still show better performance without requiring an exceeding amount of function evaluations.

Large-W limit of the knapsack problem.

We formulate the knapsack problem (KP) as a statistical physics system and compute the corresponding partition function as an integral in the complex plane. The introduced formalism allows us to derive three statistical-physics-based algorithms for the KP: one based on the recursive definition of the exact partition function, another based on the large weight limit of that partition function, and a final one based on the zero-temperature limit of the second. Comparing the performances of the algorithms, we find that they do not consistently outperform (in terms of runtime and accuracy) dynamic programming, annealing, or standard greedy algorithms. However, the exact partition function is shown to reproduce the dynamic programming solution to the KP, and the zero-temperature algorithm is shown to produce a greedy solution. Therefore, although dynamic programming and greedy solutions to the KP are conceptually distinct, a statistical physics formalism introduced reveals that the large weight-constraint limit of the former leads to the latter. We conclude by discussing how to extend this formalism in order to obtain more accurate versions of the introduced algorithms and other similar combinatorial optimization problems.

Two-Stage Probe-Based Search Optimization Algorithm for the Traveling Salesman Problems

As a classical combinatorial optimization problem, the traveling salesman problem (TSP) has been extensively investigated in the fields of Artificial Intelligence and Operations Research. Due to being NP-complete, it is still rather challenging to solve both effectively and efficiently. Because of its high theoretical significance and wide practical applications, great effort has been undertaken to solve it from the point of view of intelligent search. In this paper, we propose a two-stage probe-based search optimization algorithm for solving both symmetric and asymmetric TSPs through the stages of route development and a self-escape mechanism. Specifically, in the first stage, a reasonable proportion threshold filter of potential basis probes or partial routes is set up at each step during the complete route development process. In this way, the poor basis probes with longer routes are filtered out automatically. Moreover, four local augmentation operators are further employed to improve these potential basis probes at each step. In the second stage, a self-escape mechanism or operation is further implemented on the obtained complete routes to prevent the probe-based search from being trapped in a locally optimal solution. The experimental results on a collection of benchmark TSP datasets demonstrate that our proposed algorithm is more effective than other state-of-the-art optimization algorithms. In fact, it achieves the best-known TSP benchmark solutions in many datasets, while, in certain cases, it even generates solutions that are better than the best-known TSP benchmark solutions.