Showing papers on "Heuristic published in 2018"

Link prediction based on graph neural networks

Abstract: Link prediction is a key problem for network-structured data. Link prediction heuristics use some score functions, such as common neighbors and Katz index, to measure the likelihood of links. They have obtained wide practical uses due to their simplicity, interpretability, and for some of them, scalability. However, every heuristic has a strong assumption on when two nodes are likely to link, which limits their effectiveness on networks where these assumptions fail. In this regard, a more reasonable way should be learning a suitable heuristic from a given network instead of using predefined ones. By extracting a local subgraph around each target link, we aim to learn a function mapping the subgraph patterns to link existence, thus automatically learning a "heuristic" that suits the current network. In this paper, we study this heuristic learning paradigm for link prediction. First, we develop a novel γ-decaying heuristic theory. The theory unifies a wide range of heuristics in a single framework, and proves that all these heuristics can be well approximated from local subgraphs. Our results show that local subgraphs reserve rich information related to link existence. Second, based on the γ-decaying theory, we propose a new method to learn heuristics from local subgraphs using a graph neural network (GNN). Its experimental results show unprecedented performance, working consistently well on a wide range of problems.

Deep Reinforcement Learning for Dynamic Multichannel Access in Wireless Networks

Abstract: We consider a dynamic multichannel access problem, where multiple correlated channels follow an unknown joint Markov model and users select the channel to transmit data. The objective is to find a policy that maximizes the expected long-term number of successful transmissions. The problem is formulated as a partially observable Markov decision process with unknown system dynamics. To overcome the challenges of unknown dynamics and prohibitive computation, we apply the concept of reinforcement learning and implement a deep Q-network (DQN). We first study the optimal policy for fixed-pattern channel switching with known system dynamics and show through simulations that DQN can achieve the same optimal performance without knowing the system statistics. We then compare the performance of DQN with a Myopic policy and a Whittle Index-based heuristic through both more general simulations as well as real data trace and show that DQN achieves near-optimal performance in more complex situations. Finally, we propose an adaptive DQN approach with the capability to adapt its learning in time-varying scenarios.

Combinatorial optimization with graph convolutional networks and guided tree search

Abstract: We present a learning-based approach to computing solutions for certain NP-hard problems. Our approach combines deep learning techniques with useful algorithmic elements from classic heuristics. The central component is a graph convolutional network that is trained to estimate the likelihood, for each vertex in a graph, of whether this vertex is part of the optimal solution. The network is designed and trained to synthesize a diverse set of solutions, which enables rapid exploration of the solution space via tree search. The presented approach is evaluated on four canonical NP-hard problems and five datasets, which include benchmark satisfiability problems and real social network graphs with up to a hundred thousand nodes. Experimental results demonstrate that the presented approach substantially outperforms recent deep learning work, and performs on par with highly optimized state-of-the-art heuristic solvers for some NP-hard problems. Experiments indicate that our approach generalizes across datasets, and scales to graphs that are orders of magnitude larger than those used during training.

Link Prediction Based on Graph Neural Networks

Abstract: Link prediction is a key problem for network-structured data. Link prediction heuristics use some score functions, such as common neighbors and Katz index, to measure the likelihood of links. They have obtained wide practical uses due to their simplicity, interpretability, and for some of them, scalability. However, every heuristic has a strong assumption on when two nodes are likely to link, which limits their effectiveness on networks where these assumptions fail. In this regard, a more reasonable way should be learning a suitable heuristic from a given network instead of using predefined ones. By extracting a local subgraph around each target link, we aim to learn a function mapping the subgraph patterns to link existence, thus automatically learning a `heuristic' that suits the current network. In this paper, we study this heuristic learning paradigm for link prediction. First, we develop a novel $\gamma$-decaying heuristic theory. The theory unifies a wide range of heuristics in a single framework, and proves that all these heuristics can be well approximated from local subgraphs. Our results show that local subgraphs reserve rich information related to link existence. Second, based on the $\gamma$-decaying theory, we propose a new algorithm to learn heuristics from local subgraphs using a graph neural network (GNN). Its experimental results show unprecedented performance, working consistently well on a wide range of problems.

Learning Heuristics for the TSP by Policy Gradient

Abstract: The aim of the study is to provide interesting insights on how efficient machine learning algorithms could be adapted to solve combinatorial optimization problems in conjunction with existing heuristic procedures. More specifically, we extend the neural combinatorial optimization framework to solve the traveling salesman problem (TSP). In this framework, the city coordinates are used as inputs and the neural network is trained using reinforcement learning to predict a distribution over city permutations. Our proposed framework differs from the one in [1] since we do not make use of the Long Short-Term Memory (LSTM) architecture and we opted to design our own critic to compute a baseline for the tour length which results in more efficient learning. More importantly, we further enhance the solution approach with the well-known 2-opt heuristic. The results show that the performance of the proposed framework alone is generally as good as high performance heuristics (OR-Tools). When the framework is equipped with a simple 2-opt procedure, it could outperform such heuristics and achieve close to optimal results on 2D Euclidean graphs. This demonstrates that our approach based on machine learning techniques could learn good heuristics which, once being enhanced with a simple local search, yield promising results.

Fractional Programming for Communication Systems—Part II: Uplink Scheduling via Matching

Abstract: This two-part paper develops novel methodologies for using fractional programming (FP) techniques to design and optimize communication systems. Part I of this paper proposes a new quadratic transform for FP and treats its application for continuous optimization problems. In this Part II of the paper, we study discrete problems, such as those involving user scheduling, which are considerably more difficult to solve. Unlike the continuous problems, discrete or mixed discrete-continuous problems normally cannot be recast as convex problems. In contrast to the common heuristic of relaxing the discrete variables, this work reformulates the original problem in an FP form amenable to distributed combinatorial optimization. The paper illustrates this methodology by tackling the important and challenging problem of uplink coordinated multicell user scheduling in wireless cellular systems. Uplink scheduling is more challenging than downlink scheduling, because uplink user scheduling decisions significantly affect the interference pattern in nearby cells. Furthermore, the discrete scheduling variable needs to be optimized jointly with continuous variables such as transmit power levels and beamformers. The main idea of the proposed FP approach is to decouple the interaction among the interfering links, thereby permitting a distributed and joint optimization of the discrete and continuous variables with provable convergence. The paper shows that the well-known weighted minimum mean-square-error (WMMSE) algorithm can also be derived from a particular use of FP; but our proposed FP-based method significantly outperforms WMMSE when discrete user scheduling variables are involved, both in term of run-time efficiency and optimizing results.

Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices

Abstract: The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems. Despite its promise for near-term quantum applications, not much is currently understood about QAOA's performance beyond its lowest-depth variant. An essential but missing ingredient for understanding and deploying QAOA is a constructive approach to carry out the outer-loop classical optimization. We provide an in-depth study of the performance of QAOA on MaxCut problems by developing an efficient parameter-optimization procedure and revealing its ability to exploit non-adiabatic operations. Building on observed patterns in optimal parameters, we propose heuristic strategies for initializing optimizations to find quasi-optimal $p$-level QAOA parameters in $O(\text{poly}(p))$ time, whereas the standard strategy of random initialization requires $2^{O(p)}$ optimization runs to achieve similar performance. We then benchmark QAOA and compare it with quantum annealing, especially on difficult instances where adiabatic quantum annealing fails due to small spectral gaps. The comparison reveals that QAOA can learn via optimization to utilize non-adiabatic mechanisms to circumvent the challenges associated with vanishing spectral gaps. Finally, we provide a realistic resource analysis on the experimental implementation of QAOA. When quantum fluctuations in measurements are accounted for, we illustrate that optimization will be important only for problem sizes beyond numerical simulations, but accessible on near-term devices. We propose a feasible implementation of large MaxCut problems with a few hundred vertices in a system of 2D neutral atoms, reaching the regime to challenge the best classical algorithms.

Automated Driving in Uncertain Environments: Planning With Interaction and Uncertain Maneuver Prediction

Abstract: Automated driving requires decision making in dynamic and uncertain environments. The uncertainty from the prediction originates from the noisy sensor data and from the fact that the intention of human drivers cannot be directly measured. This problem is formulated as a partially observable Markov decision process (POMDP) with the intended route of the other vehicles as hidden variables. The solution of the POMDP is a policy determining the optimal acceleration of the ego vehicle along a preplanned path. Therefore, the policy is optimized for the most likely future scenarios resulting from an interactive, probabilistic motion model for the other vehicles. Considering possible future measurements of the surrounding cars allows the autonomous car to incorporate the estimated change in future prediction accuracy in the optimal policy. A compact representation results in a low-dimensional state-space. Thus, the problem can be solved online for varying road layouts and number of vehicles. This is done with a point-based solver in an anytime fashion on a continuous state-space. Our evaluation is threefold: At first, the convergence of the algorithm is evaluated and it is shown how the convergence can be improved with an additional search heuristic. Second, we show various planning scenarios to demonstrate how the introduction of different considered uncertainties results in more conservative planning. At the end, we show online simulations for the crossing of complex (unsignalized) intersections. We can demonstrate that our approach performs nearly as good as with full prior information about the intentions of the other vehicles and clearly outperforms reactive approaches.

Non-convex Optimization for Machine Learning

Abstract: A vast majority of machine learning algorithms train their models andperform inference by solving optimization problems. In order to capturethe learning and prediction problems accurately, structural constraintssuch as sparsity or low rank are frequently imposed or else the objectiveitself is designed to be a non-convex function. This is especially trueof algorithms that operate in high-dimensional spaces or that trainnon-linear models such as tensor models and deep networks.The freedom to express the learning problem as a non-convex optimizationproblem gives immense modeling power to the algorithmdesigner, but often such problems are NP-hard to solve. A popularworkaround to this has been to relax non-convex problems to convexones and use traditional methods to solve the convex relaxed optimizationproblems. However this approach may be lossy and neverthelesspresents significant challenges for large scale optimization.On the other hand, direct approaches to non-convex optimizationhave met with resounding success in several domains and remain themethods of choice for the practitioner, as they frequently outperformrelaxation-based techniques - popular heuristics include projected gradientdescent and alternating minimization. However, these are oftenpoorly understood in terms of their convergence and other properties.This monograph presents a selection of recent advances that bridgea long-standing gap in our understanding of these heuristics. We hopethat an insight into the inner workings of these methods will allow thereader to appreciate the unique marriage of task structure and generativemodels that allow these heuristic techniques to provably succeed.The monograph will lead the reader through several widely used nonconvexoptimization techniques, as well as applications thereof. Thegoal of this monograph is to both, introduce the rich literature in thisarea, as well as equip the reader with the tools and techniques neededto analyze these simple procedures for non-convex problems.

Learning to Optimize Join Queries With Deep Reinforcement Learning.

Abstract: Exhaustive enumeration of all possible join orders is often avoided, and most optimizers leverage heuristics to prune the search space. The design and implementation of heuristics are well-understood when the cost model is roughly linear, and we find that these heuristics can be significantly suboptimal when there are non-linearities in cost. Ideally, instead of a fixed heuristic, we would want a strategy to guide the search space in a more data-driven way---tailoring the search to a specific dataset and query workload. Recognizing the link between classical Dynamic Programming enumeration methods and recent results in Reinforcement Learning (RL), we propose a new method for learning optimized join search strategies. We present our RL-based DQ optimizer, which currently optimizes select-project-join blocks. We implement three versions of DQ to illustrate the ease of integration into existing DBMSes: (1) A version built on top of Apache Calcite, (2) a version integrated into PostgreSQL, and (3) a version integrated into SparkSQL. Our extensive evaluation shows that DQ achieves plans with optimization costs and query execution times competitive with the native query optimizer in each system, but can execute significantly faster after learning (often by orders of magnitude).

Deep Reinforcement Learning for Interference-Aware Path Planning of Cellular-Connected UAVs

Abstract: In this paper, an interference-aware path planning scheme for a network of cellular-connected unmanned aerial vehicles (UAVs) is proposed. In particular, each UAV acts as a cellular user equipment (UE) and aims at achieving a tradeoff between maximizing energy efficiency and minimizing both wireless latency and the interference caused on the ground network along its path. The problem is cast as a dynamic game among UAVs. To solve this game, a deep reinforcement learning algorithm, based on echo state network (ESN) cells, is proposed. The introduced deep ESN architecture is trained to allow each UAV to map each observation of the network state to an action, with the goal of minimizing a sequence of time-dependent utility functions. Each UAV uses ESN to learn its optimal path, transmission power, and cell association vector at different locations along its path. The proposed algorithm is shown to reach a subgame perfect Nash equilibrium upon convergence. Simulation results show that the proposed scheme achieves better wireless latency per UAV and rate per ground UE while requiring a number of steps that is comparable to a heuristic baseline that considers moving via the shortest distance towards the corresponding destinations.

Boolean Decision Rules via Column Generation

Abstract: This paper considers the learning of Boolean rules in either disjunctive normal form (DNF, OR-of-ANDs, equivalent to decision rule sets) or conjunctive normal form (CNF, AND-of-ORs) as an interpretable model for classification. An integer program is formulated to optimally trade classification accuracy for rule simplicity. Column generation (CG) is used to efficiently search over an exponential number of candidate clauses (conjunctions or disjunctions) without the need for heuristic rule mining. This approach also bounds the gap between the selected rule set and the best possible rule set on the training data. To handle large datasets, we propose an approximate CG algorithm using randomization. Compared to three recently proposed alternatives, the CG algorithm dominates the accuracy-simplicity trade-off in 8 out of 16 datasets. When maximized for accuracy, CG is competitive with rule learners designed for this purpose, sometimes finding significantly simpler solutions that are no less accurate.

A parallel cooperative hybrid method based on ant colony optimization and 3-Opt algorithm for solving traveling salesman problem

Abstract: This article presented a parallel cooperative hybrid algorithm for solving traveling salesman problem. Although heuristic approaches and hybrid methods obtain good results in solving the TSP, they cannot successfully avoid getting stuck to local optima. Furthermore, their processing duration unluckily takes a long time. To overcome these deficiencies, we propose the parallel cooperative hybrid algorithm (PACO-3Opt) based on ant colony optimization. This method uses the 3-Opt algorithm to avoid local minima. PACO-3Opt has multiple colonies and a master---slave paradigm. Each colony runs ACO to generate the solutions. After a predefined number of iterations, each colony primarily runs 3-Opt to improve the solutions and then shares the best tour with other colonies. This process continues until the termination criterion meets. Thus, it can reach the global optimum. PACO-3Opt was compared with previous algorithms in the literature. The experimental results show that PACO-3Opt is more efficient and reliable than the other algorithms.

DeepType: Multilingual Entity Linking by Neural Type System Evolution

Abstract: The wealth of structured (e.g. Wikidata) and unstructured data about the world available today presents an incredible opportunity for tomorrow's Artificial Intelligence. So far, integration of these two different modalities is a difficult process, involving many decisions concerning how best to represent the information so that it will be captured or useful, and hand-labeling large amounts of data. DeepType overcomes this challenge by explicitly integrating symbolic information into the reasoning process of a neural network with a type system. First we construct a type system, and second, we use it to constrain the outputs of a neural network to respect the symbolic structure. We achieve this by reformulating the design problem into a mixed integer problem: create a type system and subsequently train a neural network with it. In this reformulation discrete variables select which parent-child relations from an ontology are types within the type system, while continuous variables control a classifier fit to the type system. The original problem cannot be solved exactly, so we propose a 2-step algorithm: 1) heuristic search or stochastic optimization over discrete variables that define a type system informed by an Oracle and a Learnability heuristic, 2) gradient descent to fit classifier parameters. We apply DeepType to the problem of Entity Linking on three standard datasets (i.e. WikiDisamb30, CoNLL (YAGO), TAC KBP 2010) and find that it outperforms all existing solutions by a wide margin, including approaches that rely on a human-designed type system or recent deep learning-based entity embeddings, while explicitly using symbolic information lets it integrate new entities without retraining.

A Collaborative Multiobjective Fruit Fly Optimization Algorithm for the Resource Constrained Unrelated Parallel Machine Green Scheduling Problem

Abstract: Due to the development of the green economy, green manufacturing has been a hot topic. This paper proposes a new problem, i.e., the resource constrained unrelated parallel machine green manufacturing scheduling problem (RCUPMGSP) with the criteria of minimizing the makespan and the total carbon emission. To solve the problem, a collaborative multiobjective fruit fly optimization algorithm (CMFOA) is proposed. First, a job-speed pair-based solution representation is presented, and an effective decoding method is designed. Second, a heuristic for initialization of the population is proposed. Third, three collaborative search operators are designed to handle three subproblems in the smell-based search phase, i.e., job-to-machine assignment, job sequence, and processing speed selection. The technique for order preference by similarity to an ideal solution and the fast nondominated sorting methods are both employed for multiobjective evaluation in the vision-based search phase. Moreover, a critical-path-based carbon saving technique is designed according to the problem analysis to further improve the nondominated solutions explored in the fruit fly optimization algorithm-based evolution. In addition, the effect of parameter setting is investigated and the suitable parameter values are recommended. Finally, numerical tests and comparisons are carried out using the randomly generated instances, which show that the CMFOA is able to obtain more and better nondominated solutions than other algorithms. The comparisons also demonstrate the effectiveness of the collaborative scheme and the carbon saving technique as well as the CMFOA in solving the RCUPMGSP.

DeepType: Multilingual Entity Linking by Neural Type System Evolution

Abstract: The wealth of structured (e.g. Wikidata) and unstructured data about the world available today presents an incredible opportunity for tomorrow's Artificial Intelligence. So far, integration of these two different modalities is a difficult process, involving many decisions concerning how best to represent the information so that it will be captured or useful, and hand-labeling large amounts of data. DeepType overcomes this challenge by explicitly integrating symbolic information into the reasoning process of a neural network with a type system. First we construct a type system, and second, we use it to constrain the outputs of a neural network to respect the symbolic structure. We achieve this by reformulating the design problem into a mixed integer problem: create a type system and subsequently train a neural network with it. In this reformulation discrete variables select which parent-child relations from an ontology are types within the type system, while continuous variables control a classifier fit to the type system. The original problem cannot be solved exactly, so we propose a 2-step algorithm: 1) heuristic search or stochastic optimization over discrete variables that define a type system informed by an Oracle and a Learnability heuristic, 2) gradient descent to fit classifier parameters. We apply DeepType to the problem of Entity Linking on three standard datasets (i.e. WikiDisamb30, CoNLL (YAGO), TAC KBP 2010) and find that it outperforms all existing solutions by a wide margin, including approaches that rely on a human-designed type system or recent deep learning-based entity embeddings, while explicitly using symbolic information lets it integrate new entities without retraining.

Query optimization through the looking glass, and what we found running the Join Order Benchmark

Abstract: Finding a good join order is crucial for query performance. In this paper, we introduce the Join Order Benchmark that works on real-life data riddled with correlations and introduces 113 complex join queries. We experimentally revisit the main components in the classic query optimizer architecture using a complex, real-world data set and realistic multi-join queries. For this purpose, we describe cardinality-estimate injection and extraction techniques that allow us to compare the cardinality estimators of multiple industrial SQL implementations on equal footing, and to characterize the value of having perfect cardinality estimates. Our investigation shows that all industrial-strength cardinality estimators routinely produce large errors: though cardinality estimation using table samples solves the problem for single-table queries, there are still no techniques in industrial systems that can deal accurately with join-crossing correlated query predicates. We further show that while estimates are essential for finding a good join order, query performance is unsatisfactory if the query engine relies too heavily on these estimates. Using another set of experiments that measure the impact of the cost model, we find that it has much less influence on query performance than the cardinality estimates. We investigate plan enumeration techniques comparing exhaustive dynamic programming with heuristic algorithms and find that exhaustive enumeration improves performance despite the suboptimal cardinality estimates. Finally, we extend our investigation from main-memory only, to also include disk-based query processing. Here, we find that though accurate cardinality estimation should be the first priority, other aspects such as modeling random versus sequential I/O are also important to predict query runtime.

Urban freight transport using passenger rail network: Scientific issues and quantitative analysis

Abstract: This paper addresses a real-life problem arising in the ongoing “Grand Paris” project. We investigate an environment-friendly urban freight transportation alternative using passenger rail network, by providing a decision support tool for decision makers to assess the technical feasibility, the impact on services to passengers, the needs in infrastructure and hence in investment. We identify relevant scientific issues that need to be addressed in this topic at strategical, tactical and operational levels. Then we focus on the Freight-Rail-Transport-Scheduling Problem which provides valuable information to and constitutes a basis for other related problems. This problem is first formulated into a MIP. We prove its NP-hardness and hence propose a heuristic based on dispatching rules and a single-train-based decomposition heuristic. The performances of these heuristics are evaluated via employing a discrete-event simulation approach, which also provides a general framework which supports decision-makers in modelling and evaluating the dynamics of such a system for various alternative solutions under various scenarios.

A Hybrid Genetic Algorithm for Multi-Trip Green Capacitated Arc Routing Problem in the Scope of Urban Services

Abstract: Greenhouse gases (GHG) are the main reason for the global warming during the past decades. On the other hand, establishing a well-structured transportation system will yield to create least cost-pollution. This paper addresses a novel model for the multi-trip Green Capacitated Arc Routing Problem (G-CARP) with the aim of minimizing total cost including the cost of generation and emission of greenhouse gases, the cost of vehicle usage and routing cost. The cost of generation and emission of greenhouse gases is based on the calculation of the amount of carbon dioxide emitted from vehicles, which depends on such factors as the vehicle speed, weather conditions, load on the vehicle and traveled distance. The main applications of this problem are in municipalities for urban waste collection, road surface marking and so forth. Due to NP-hardness of the problem, a Hybrid Genetic Algorithm (HGA) is developed, wherein a heuristic and simulated annealing algorithm are applied to generate initial solutions and a Genetic Algorithm (GA) is then used to generate the best possible solution. The obtained numerical results indicate that the proposed algorithm could present desirable performance within a suitable computational run time. Finally, a sensitivity analysis is implemented on the maximum available time of the vehicles in order to determine the optimal policy.

Joint Dynamic Pricing and Order Fulfillment for E-commerce Retailers

Abstract: We consider an e-commerce retailer (e-tailer) who sells a catalog of products to customers from different regions during a finite selling season and fulfills orders through multiple fulfillment centers. The e-tailer faces a joint pricing and fulfillment (JPF) optimization problem: at the beginning of each period, the e-tailer needs to jointly decide the price for each product and how to fulfill an incoming order (i.e., from which warehouse to ship the order). The objective of the e-tailer is to maximize its total expected profits defined as total expected revenues minus total expected shipping costs. (All other costs are fixed in this problem.) The exact optimal policy for JPF is difficult to solve; so, we propose two heuristic controls that have provably good performance compared to reasonable benchmarks. Our first heuristic control directly uses the solution of a deterministic approximation of JPF as its control parameters. Our second heuristic control improves the first one by adaptively adjusting the ...

Taming VAEs.

Abstract: In spite of remarkable progress in deep latent variable generative modeling, training still remains a challenge due to a combination of optimization and generalization issues. In practice, a combination of heuristic algorithms (such as hand-crafted annealing of KL-terms) is often used in order to achieve the desired results, but such solutions are not robust to changes in model architecture or dataset. The best settings can often vary dramatically from one problem to another, which requires doing expensive parameter sweeps for each new case. Here we develop on the idea of training VAEs with additional constraints as a way to control their behaviour. We first present a detailed theoretical analysis of constrained VAEs, expanding our understanding of how these models work. We then introduce and analyze a practical algorithm termed Generalized ELBO with Constrained Optimization, GECO. The main advantage of GECO for the machine learning practitioner is a more intuitive, yet principled, process of tuning the loss. This involves defining of a set of constraints, which typically have an explicit relation to the desired model performance, in contrast to tweaking abstract hyper-parameters which implicitly affect the model behavior. Encouraging experimental results in several standard datasets indicate that GECO is a very robust and effective tool to balance reconstruction and compression constraints.

A hybrid algorithm using a genetic algorithm and multiagent reinforcement learning heuristic to solve the traveling salesman problem

Abstract: In recent years, hybrid genetic algorithms (GAs) have received significant interest and are widely being used to solve real-world problems. The hybridization of heuristic methods aims at incorporating benefits of stand-alone heuristics in order to achieve better results for the optimization problem. In this paper, we propose a hybridization of GAs and Multiagent Reinforcement Learning (MARL) heuristic for solving Traveling Salesman Problem (TSP). The hybridization process is implemented by producing the initial population of GA, using MARL heuristic. In this way, GA with a novel crossover operator, which we have called Smart Multi-point crossover, acts as tour improvement heuristic and MARL acts as construction heuristic. Numerical results based on several TSP datasets taken from the TSPLIB demonstrate that proposed method found optimum solution of many TSP datasets and near optimum of the others and enable to compete with nine state-of-the-art algorithms, in terms of solution quality and CPU time.