Traveling Salesman Problem Solver Research Articles

A novel memetic algorithm for solving the generalized traveling salesman problem

Abstract This paper investigates the Generalized Traveling Salesman Problem (GTSP), which is an extension of the well-known Traveling Salesman Problem (TSP), and it searches for an optimal tour in a clustered graph, such that every cluster is visited exactly once. In this paper, we describe a novel Memetic Algorithm (MA) for solving efficiently the GTSP. Our proposed MA has at its core a genetic algorithm (GA), completed by a Chromosome Enhancement Procedure (CEP), which is based on a TSP solver and the Shortest Path (SP) algorithm and for improving the convergence characteristics of the GA, a Local Search (LS) operation is applied for the best chromosomes in each generation. We tested our algorithm on a set of well-known instances from the literature and the achieved results prove that our novel memetic algorithm is highly competitive against the existing solution approaches from the specialized literature.

An edge-aware graph autoencoder trained on scale-imbalanced data for traveling salesman problems

In recent years, there has been a notable surge in research on machine learning techniques for combinatorial optimization. It has been shown that learning-based methods outperform traditional heuristics and mathematical solvers on the Traveling Salesman Problem (TSP) in terms of both performance and computational efficiency. However, most learning-based TSP solvers are primarily designed for fixed-scale TSP instances, and also require a large number of training samples to achieve optimal performance. To fill this gap, this work proposes a data-driven graph representation learning method for solving TSPs with various numbers of cities. Specifically, we formulate the TSP as a link prediction task and propose an edge-aware graph autoencoder (EdgeGAE) model that can solve TSPs by learning from various-scale samples with an imbalanced distribution. A residual gated encoder is trained to learn latent edge embeddings, followed by an edge-centered decoder to output link predictions in an end-to-end manner. Furthermore, we introduce an active sampling strategy into the training process to improve the model’s generalization capability in large-scale scenarios. To investigate the model’s practical applicability, we generate a scale-imbalanced dataset comprising 50,000 TSP instances ranging from 50 to 500 cities. The experimental results demonstrate that the proposed edge-aware graph autoencoder model achieves a highly competitive performance among state-of-the-art graph learning-based approaches in solving TSPs with various scales, implying its remarkable potential in dealing with practical optimization challenges.

Neural TSP Solver with Progressive Distillation

Travelling salesman problem (TSP) is NP-Hard with exponential search space. Recently, the adoption of encoder-decoder models as neural TSP solvers has emerged as an attractive topic because they can instantly obtain near-optimal results for small-scale instances. Nevertheless, their training efficiency and solution quality degrade dramatically when dealing with large-scale problems. To address the issue, we propose a novel progressive distillation framework, by adopting curriculum learning to train TSP samples in increasing order of their problem size and progressively distilling high-level knowledge from small models to large models via a distillation loss. In other words, the trained small models are used as the teacher network to guide action selection when training large models. To accelerate training speed, we also propose a Delaunary-graph based action mask and a new attention-based decoder to reduce decoding cost. Experimental results show that our approach establishes clear advantages over existing encoder-decoder models in terms of training effectiveness and solution quality. In addition, we validate its usefulness as an initial solution generator for the state-of-the-art TSP solvers, whose probability of obtaining the optimal solution can be further improved in such a hybrid manner.

IEDW: A BIM-based indoor electric distribution wiring algorithm using graph theory and capacity-limited multiple traveling salesman problem solver

Building Information Modeling (BIM) has emerged as a new paradigm for the construction industry. The combination of electric distribution systems design and BIM is the future development trend of architectural design. The traditional electric distribution wiring design based on BIM is mainly carried out manually, which is laborious and error-prone. To cope with this challenge, some studies have proposed explorations of automated BIM wiring design based on graphs in building structures, HAVC, and AEC fields, but few efforts were made on automatic electric distribution wiring. In view of its complicated nature, actual construction factors such as design specifications, wiring costs, etc. should be considered with electric distribution wiring simultaneously. To address this issue, this study proposes a BIM-based automatic indoor electric distribution wiring (IEDW) scheme using graph theory and capacity-limited multiple traveling salesman problem (CMTSP). Firstly, the IEDW sets up the model for design specifications and actual wiring costs with four constraints of specification, cost, installation, and load. Then, a electric distribution wiring heterogeneous graph model is designed to solve both specification and installation constraints. Thirdly, a CMTSP-based algorithm is designed to solve the problem of automatic electric distribution wiring under cost and load constraints. Finally, the performance of IEDW is evaluated on three types of buildings. The experimental results proved that the IEDW can provide a more reasonable wiring solution than manual wiring in terms of circuit balance and total wiring length. Moreover, the IEDW can reduce the wiring cost by 8.8% on average. The proposed IEDW will enhance the efficiency of BIM-based electric systems design and delivery.

On the generation of metric TSP instances with a large integrality gap by branch-and-cut

This paper introduces a computational method for generating metric Travelling Salesman Problem (TSP) instances having a large integrality gap. The method is based on the solution of an integer programming problem, called IH-OPT, that takes as input a fractional solution of the Subtour Elimination Problem (SEP) on a TSP instance and computes a TSP instance having an integrality gap larger than or equal to the integrality gap of the first instance. The decision variables of IH-OPT are the entries of the TSP cost matrix, and the constraints are defined by the intersection of the metric cone with an exponential number of inequalities, one for each possible TSP tour. Given the very large number of constraints, we have implemented a branch-and-cut algorithm for solving IH-OPT. Then, by sampling cost vectors over the metric polytope and by solving the corresponding SEP, we can generate random fractional vertices of the SEP polytope. If we solve the IH-OPT problem for every sampled vertex using our branch-and-cut algorithm, we can select the generated TSP instance (i.e., cost vector), yielding the longest runtime for Concorde, the state-of-the-art TSP solver. Our computational results show that our method is very effective in producing challenging instances. As a by-product, we release the Hard-TSPLIB, a library of 41 small metric TSP instances which have a large integrality gap and are challenging in terms of runtime for Concorde.

A Grid-Based Two-Stage Parallel Matching Framework for Bi-Objective Euclidean Traveling Salesman Problem

Traveling salesman problem (TSP) is one of the most studied combinatorial optimization problems; several exact, heuristic or even learning-based strategies have been proposed to solve this challenging issue. Targeting on the research problem of bi-objective non-monotonic Euclidean TSP and based on the concept of the multi-agent-based approach, we propose a two-stage parallel matching approaching for solving TSP. Acting as a divide-and-conquer strategy, the merit lies in the simultaneously clustering and routing in the dividing process. Precisely, we first propose the Two-Stage Parallel Matching algorithm (TSPM) to deal with the bi-objective TSP. We then formulate the Grid-Based Two-Stage Parallel Matching (GRAPE) framework, which can synergize with TSPM, exact method, or other state-of-the-art TSP solvers, for solving large-scale Euclidean TSP. According to this framework, the original problem space is divided into smaller regions and then computed in parallel, which helps to tackle and derive solutions for larger-scale Euclidean TSP within reasonable computational resources. Preliminary evaluation based on TSPLIB testbed shows that our proposed GRAPE framework holds a decent quality of solutions in especially runtime for large-scale Euclidean TSP. Meanwhile, experiments conducted on two real-world datasets demonstrate the efficacy and adaptability of our proposed TSPM in solving the bi-objective non-monotonic TSP.

Utilization of a TSP solver for generating non-retractable, direction favouring toolpath for additive manufacturing

Toolpath planning is a crucial part of Additive Manufacturing (AM) . It significantly affects almost every aspect of part fabrication, from part properties and dimensional accuracy to print cost and time. This works implements a Travelling Salesman Problem (TSP) based algorithm to generate a toolpath for efficient and accurate area filling with a minimum number of tool retractions and 'number of turns.' The toolpath generation process starts by first digitizing the layer being processed, i.e., given a geometric contour, a series of grid points (analogous to cities) within the boundary is generated. A deposition head sequentially traverses these grid points, filling the entire area with no voids (or porosity). A detailed investigation of the digitization process, exploring the rectangular and circular array of grid points, has been presented, which further elaborates the effect of digitization on the toolpath. A novel method of reducing the number of turns in a toolpath by appropriately positioning the grid points is highlighted in the present work. The proposed path planning method is capable of orienting the toolpath such that the deposited track (or the scan lines) favors one direction of motion over the other. Finally, an account of the successful implementation of the proposed strategy in an FDM and WAAM-based printer is also presented as a part of the study.

The Circlet Inequalities: A New, Circulant-Based, Facet-Defining Inequality for the TSP

Facet-defining inequalities of the symmetric traveling salesman problem (TSP) polytope play a prominent role in both polyhedral TSP research and state-of-the-art TSP solvers. In this paper, we introduce a new class of facet-defining inequalities, the circlet inequalities. These inequalities were first conjectured in Gutekunst and Williamson [Gutekunst SC, Williamson DP (2019) Characterizing the integrality gap of the subtour LP for the circulant traveling salesman problem. SIAM J. Discrete Math. 33(4):2452–2478] when studying the circulant TSP, and they provide a bridge between polyhedral TSP research and number-theoretic investigations of Hamiltonian cycles stemming from a conjecture from Marco Buratti in 2007. The circlet inequalities exhibit circulant symmetry by placing the same weight on all edges of a given length; our main proof exploits this symmetry to prove the validity of the circlet inequalities. We then show that the circlet inequalities are facet-defining and compute their strength following Goemans [Goemans MX (1995) Worst-case comparison of valid inequalities for the TSP. Math. Programming 69:335–349]; they achieve the same worst case strength as the similarly circulant crown inequalities of Naddef and Rinaldi [Naddef D, Rinaldi G (1992) The crown inequalities for the symmetric traveling salesman polytope. Math. Oper. Res. 17(2):308–326] but are generally stronger. Funding: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program [Grant DGE-1650441] and by the National Science Foundation Division of Computing and Communications Foundations [Grant CCF-1908517].

Learning to Solve Travelling Salesman Problem with Hardness-Adaptive Curriculum

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.

Solving the clustered traveling salesman problem via traveling salesman problem methods.

The Clustered Traveling Salesman Problem (CTSP) is a variant of the popular Traveling Salesman Problem (TSP) arising from a number of real-life applications. In this work, we explore a transformation approach that solves the CTSP by converting it to the well-studied TSP. For this purpose, we first investigate a technique to convert a CTSP instance to a TSP and then apply powerful TSP solvers (including exact and heuristic solvers) to solve the resulting TSP instance. We want to answer the following questions: How do state-of-the-art TSP solvers perform on clustered instances converted from the CTSP? Do state-of-the-art TSP solvers compete well with the best performing methods specifically designed for the CTSP? For this purpose, we present intensive computational experiments on various benchmark instances to draw conclusions.

Integrated photonic traveling salesman problem probabilistic solver with polynomial calculation time complexity

The traveling salesman problem (TSP) is an NP hard problem, where the solution time becomes unrealistically large as the number of points increases. We propose and simulate a photonic integrated circuit-based TSP solver which uses the delay time of light to express the delay caused by traveling between the points. The solver is comprised of semiconductor optical amplifiers and tunable delay lines, therefore it is reconfigurable. A probabilistic algorithm is used, which enables polynomial calculation and footprint complexity with an error rate of less than 1e-12. The calculation time is 388 ms and footprint is within 10 cm squared for a TSP of 100 points.

Traveling Transporter Problem: Arranging a New Circular Route in a Public Transportation System Based on Heterogeneous Non-Monotonic Urban Data

Hybrid computational intelligent systems that synergize learning-based inference models and route planning strategies have thrived in recent years. In this article, we focus on the non-monotonicity originated from heterogeneous urban data, as well as heuristics based on neural networks, and thereafter formulate the traveling transporter problem (TTP). TTP is a multi-criteria optimization problem and may be applied to the circular route deployment in public transportation. In particular, TTP aims to find an optimized route that maximizes passenger flow according to a neural-network-based inference model and minimizes the length of the route given several constraints, including must-visit stations and the requirement for additional ones. As a variation of the traveling salesman problem (TSP), we propose a framework that first recommends new stations’ location while considering the herding effect between stations, and thereafter combines state-of-the-art TSP solvers and a metaheuristic named Trembling Hand , which is inspired by self-efficacy for solving TTP. Precisely, the proposed Trembling Hand enhances the spatial exploration considering the structural patterns, previous actions, and aging factors. Evaluation conducted on two real-world mass transit systems, Tainan and Chicago, shows that the proposed framework can outperform other state-of-the-art methods by securing the Pareto-optimal toward the objectives of TTP among comparative methods under various constrained settings.

Exponential neighborhood search for consecutive block minimization

AbstractGiven binary matrix , the term refers to the matrix obtained by permuting the columns of according to permutation . A 1‐block in a binary matrix is any maximal sequence of consecutive 1s situated on the same row. Given binary matrix , consecutive block minimization (CBM) is a combinatorial optimization problem that seeks a permutation of columns such that the number of 1‐blocks in is minimum. Since CBM is NP‐hard, we solve it by iterating local search where the neighborhood is exponential in the instance size and is always constructed in the vicinity of the best current solution. Seeking a local optimum in the exponential neighborhood amounts to solving a small traveling salesman problem (TSP). Instead, we obtain a good near local optimal solution using an implementation of Lin–Kernighan heuristic. Our method is compared with a recently published iterated local search metaheuristic as well as with two TSP solvers. The comparison shows that the proposed method provides better quality solutions.

A new global toolpath linking algorithm for different subregions with Travelling Saleman problem solver

ABSTRACT In CNC toolpath generation process, the operation of linking toolpaths from different sub-machining regions is common and inevitable. Apparently, the jumping toolpath between machining regions is invalid. They do not contribute to the machining process but only waste valuable manufacturing time; therefore, these toolpaths should be as short as possible. Many methods have been used to link toolpaths, such as the Genetic Algorithm (GA), the Particle Swarm Optimization (PSO) or even the greedy algorithm. However, GA and PSO require multiple iterations to find the global optimum, while greedy algorithm selects the current shortest connection each time without considering the global optimum. To reduce the total length of non-productive toolpaths and save computing time, in this paper, a new method is proposed by modeling the toolpath linking problem purely as a traveling salesman problem (TSP). The initial toolpaths in different subregions are generated in ordinary ways. Each toolpath of a subregion has two endpoints, which can be simplified as a line segment. In this way, the toolpath linking problem can be considered as a segment TSP: finding the shortest tour through all the segments. In this paper, the efficient TSP solver using Lin-Kernighan–Helsgaun (LKH) algorithm is employed and modified for the segment TSP application. The distance function between ‘cities’ is redefined to adapt the segments TSP. Finally, the feasibility of the proposed method is verified with several examples. The comparison with the result of traditional greedy algorithm proves the superiority of the proposed method.

Complexity indices for the traveling salesman problem continued

We consider the symmetric traveling salesman problem (TSP) with instances represented by complete graphs G with distances between cities as edge weights. A complexity index is an invariant of an instance I by which we predict the execution time of an exact TSP algorithm for I. In the paper [5] we have considered some short edge subgraphs of G and defined several new invariants related to their connected components. Extensive computational experiments with instances on 50 vertices with the uniform distribution of integer edge weights in the interval [1,100] show that there exists correlation between the sequences of selected invariants and the sequence of execution times of the well-known TSP Solver Concorde. In this paper we extend these considerations for instances up to 100 vertices.

Homotopic Convex Transformation: A New Landscape Smoothing Method for the Traveling Salesman Problem.

This article proposes a novel landscape smoothing method for the symmetric traveling salesman problem (TSP). We first define the homotopic convex (HC) transformation of a TSP as a convex combination of a well-constructed simple TSP and the original TSP. The simple TSP, called the convex-hull TSP, is constructed by transforming a known local or global optimum. We observe that controlled by the coefficient of the convex combination, with local or global optimum: 1) the landscape of the HC transformed TSP is smoothed in terms that its number of local optima is reduced compared to the original TSP and 2) the fitness distance correlation of the HC transformed TSP is increased. Furthermore, we observe that the smoothing effect of the HC transformation depends highly on the quality of the used optimum. A high-quality optimum leads to a better smoothing effect than a low-quality optimum. We then propose an iterative algorithmic framework in which the proposed HC transformation is combined within a heuristic TSP solver. It works as an escaping scheme from local optima aiming to improve the global searchability of the combined heuristic. Case studies using the 3-Opt and the Lin-Kernighan local search as the heuristic solver show that the resultant algorithms significantly outperform their counterparts and two other smoothing-based TSP heuristic solvers on most of the test instances with up to 20 000 cities.

Improving ant colony optimization algorithm with epsilon greedy and Levy flight

Ant colony optimization (ACO) algorithm is a meta-heuristic and reinforcement learning algorithm, which has been widely applied to solve various optimization problems. The key to improving the performance of ACO is to effectively resolve the exploration/exploitation dilemma. Epsilon greedy is an important and widely applied policy-based exploration method in reinforcement learning and has also been employed to improve ACO algorithms as the pseudo-stochastic mechanism. Levy flight is based on Levy distribution and helps to balance searching space and speed for global optimization. Taking advantage of both epsilon greedy and Levy flight, a greedy–Levy ACO incorporating these two approaches is proposed to solve complicated combinatorial optimization problems. Specifically, it is implemented on the top of max–min ACO to solve the traveling salesman problem (TSP) problems. According to the computational experiments using standard TSPLIB instances, greedy–Levy ACO outperforms max–min ACO and other latest TSP solvers, which demonstrates the effectiveness of the proposed methodology.

An SMT-Based Approach to Motion Planning for Multiple Robots With Complex Constraints

In this paper, we propose a new method for solving multirobot motion planning problems with complex constraints. We focus on an important class of problems that require an allocation of spatially distributed tasks to robots, along with efficient paths for each robot to visits their task locations. We introduce a framework for solving these problems that naturally couples allocation with path planning. The allocation problem is encoded as a Boolean Satisfiability problem (SAT) and the path planning problem is encoded as a traveling salesman problem (TSP). In addition, the framework can handle complex constraints such as battery life limitations, robot carrying capacities, and robot-task incompatibilities. We propose an algorithm that leverages recent advances in Satisfiability Modulo Theory (SMT) to combine state-of-the-art SAT and TSP solvers. We characterize the correctness of our algorithm and evaluate it in simulation on a series of patrolling, periodic routing, and multirobot sample collection problems. The results show our algorithm significantly outperforms state-of-the-art mathematical programming solvers.

Application of decision theory and bee-inspired method to railway system route optimization

ABSTRACTRoute planning for multiple destinations via a railway system (RS) is challenging, especially in a complex network with hundreds of stations and interchanges, resulting in a railway traveling salesman problem (RTSP), which is a variant of the traveling salesman problem (TSP). Limited attention has been devoted to solving the RTSP, despite the increase in the number of people using RSs and the added complexity of network expansions. An optimization algorithm and mathematical model based on the foraging behavior of bees and decision theory were used in this paper to identify the optimum RS route to multiple destinations before returning to the first station. Data collected from RSs in Japan and Malaysia were used to create 200 test cases (100 cases with each dataset), and the results were compared to specific solutions and the official optimum route planner to prove that the model is a promising approximation method. The solutions were evaluated and verified by comparing results from another 200 brute-force cases generated by the Wiley TSP Solver. The results obtained from the experiments prove the reliability and capability of the route planning and optimization solutions for RSs with different complexities and in different environments without the assistance of information technology.

A variable neighborhood search for flying sidekick traveling salesman problem

AbstractThe efficiency and dynamism of unmanned aerial vehicles, or drones, have presented substantial application opportunities in several industries in the last years. Notably, logistic companies have given close attention to these vehicles to reduce delivery time and operational cost. A variant of the traveling salesman problem (TSP), called the flying sidekick traveling salesman problem, was introduced involving drone‐assisted parcel delivery. The drone launches from the truck, proceeds to deliver parcels to a customer, and then is recovered by the truck at a third location. While the drone travels through a trip, the truck delivers parcels to other customers as long as the drone has enough battery to hover waiting for the truck. This work proposes a hybrid heuristic where the initial solution is created from the optimal TSP solution reached by a TSP solver. Next, an implementation of the general variable neighborhood search is employed to obtain the delivery routes of truck and drone. Computational experiments show the potential of the algorithm to improve significantly delivery time. Furthermore, we provide a new set of instances based on the well‐known traveling salesman problem library instances.