Traveling Salesman Problem Tour Research Articles

Improving analytic approximations of TSP tour lengths with adjustment factors

Optimizing Traveling Salesman Problem (TSP) tours requires substantial computational effort, leading researchers to develop approximations relating tour length to the number of visited points, n. Existing models, such as the √nA predictor, effectively approximate tour lengths for large-capacity vehicles but sacrifice accuracy for small n values relevant for most practical applications. Consequently, this study addresses this gap by proposing models with uniform node distributions, which incorporate realistic factors, such as central vs. random starting points and various service zone shapes. These factors are then integrated into a single equation, enhancing applicability. Furthermore, the exponent of n is statistically estimated to be significantly different from 0.5, challenging previous studies. Our proposed model estimates TSP tour lengths more accurately, particularly for small n values, and maintains accuracy for large n values, with errors below 3.11% for up to 600 points. This model offers a more precise and versatile alternative to current models.

Hybrid Parallel Ant Colony Optimization for Application to Quantum Computing to Solve Large-Scale Combinatorial Optimization Problems

Quantum computing is a promising technology that may provide breakthrough solutions to today’s difficult problems such as breaking encryption and solving large-scale combinatorial optimization problems. An algorithm referred to as Quantum Approximate Optimization Algorithm (QAOA) has been recently proposed to approximately solve hard problems using a protocol know as bang–bang. The technique is based on unitary evolution using a Hamiltonian encoding of the objective function of the combinatorial optimization problem. The QAOA was explored in the context of several optimization problems such as the Max-Cut problem and the Traveling Salesman Problem (TSP). Due to the relatively small node-size solution capability of the available quantum computers and simulators, we developed a hybrid approach where sub-graphs of a TSP tour can be executed on a quantum computer, and the results from the quantum optimization are combined for a further optimization of the whole tour. Since the local optimization of a sub-graph is prone to becoming trapped in a local minimum, we overcame this problem by using a parallel Ant Colony Optimization (ACO) algorithm with periodic pheromone exchange between colonies. Our method exceeds existing approaches which have attempted partitioning a graph for small problems (less than 48 nodes) with sub-optimal results. We obtained optimum results for benchmarks with less than 150 nodes and results usually within 1% of the known optimal solution for benchmarks with around 2000 nodes.

Accelerate Incremental TSP Algorithms on Time Evolving Graphs with Partitioning Methods

In time-evolving graphs, the graph changes at each time interval, and the previously computed results become invalid. We addressed this issue for the traveling salesman problem (TSP) in our previous work and proposed an incremental algorithm where the TSP tour is computed from the previous result instead of the whole graph. In our current work, we have mapped the TSP problem to three partitioning methods named vertex size attribute, edge attribute, and k-means; then, we compared the TSP tour results. We have also examined the effect of increasing the number of partitions on the total computation time. Through our experiments, we have observed that the vertex size attribute performs the best because of a balanced number of vertices in each partition.

Review of Length Approximations for Tours with Few Stops

The shortest tour distance for visiting all points exactly once and returning to the origin is computed by solving the well-known traveling salesman problem (TSP). Owing to the large computational effort needed for optimizing TSP tours, researchers have developed approximations that relate the average length of TSP tours to the number of points, n, visited per tour. The existing approximations are used in transportation system planning and evaluation for estimating the distance for vehicles with large capacities (e.g., delivery trucks) where n is relatively large. However, the average TSP tour lengths would be inaccurately estimated if the approximations were derived for a wide range of n values since the tour lengths increase at a decreasing rate (i.e., with [Formula: see text]) as n increases. A comprehensive review is presented here of existing studies approximating TSP tour lengths for low n values, which have become important for some recently favored delivery alternatives (e.g., deliveries by bikes, drones, and robots).

Statistical Modeling of Quartiles, Standard Deviation, and Buffer Time Index of Optimal Tour in Traveling Salesman Problem and Implications for Travel Time Reliability

The traveling salesman problem (TSP) plays an important role in the field of transportation and logistics. While most studies focus on developing algorithms to find the shortest path and explore the average length of the shortest paths, the degree to which the shortest path deviates from its mean has not been studied. The study of deviation is important because it has implications for travel time reliability. Previous studies have used various indicators to measure this deviation, mainly including standard deviation, quantiles, and buffer time index (BTI). Therefore, this study aims to develop an empirical model to estimate the standard deviation, quantiles, and BTI for the optimal TSP tour. Experiments are performed to find the shortest path connecting N customers, which are generated randomly in a specified service area, using a genetic algorithm. The service area is a rectangle with ratio of length to width ranging from 1:1 to 8:1. Two types of lengths are considered: Euclidean and Manhattan. The number of customers considered ranges from 10 to 100 with intervals of 10. In the experimental design, the customers are generated randomly 500 times. The quartiles and standard deviations of the 500 shortest paths are recorded. The BTI is also calculated. Regression models are developed to estimate quartiles, standard deviation, and BTI using number of customers and parameters of service area as predictive variables. The models perform well on the testing data set. The constructed models can be used to estimate the standard deviation and reliability of travel time.

The balancing traveling salesman problem: application to warehouse order picking

This paper discusses the problem of predicting the length of Traveling Salesman Problem (TSP) tour in dynamic and uncertain environments. Our findings, which are guided by extensive simulation runs, provide statistical estimations for the tour length under different scenarios. One of the applications that can benefit from these estimates includes warehouse order picking, which has seen increased importance due to online shopping. The utility of statistical estimates for TSP tour length for order picking is demonstrated on a common warehouse layout.

A polynomial matrix processing heuristic algorithm for finding high quality feasible solutions for the TSP

In this paper we present a simple heuristic algorithm to find high-quality, feasible solutions, for the traveling salesman problem (TSP). We hypothesize, that the quality of the initial solution provided by the proposed heuristic will improve the performance of the subsequent algorithm in terms of number of iterations required to reach a certain level TSP solution. The proposed heuristic does not attempt to compete against known TSP algorithms and heuristics, but instead, should be considered to serve as a “pre-processor”. The method provides a simple framework for testing new node selection and neighborhood rules. The cost matrix of origin and destination pairs is processed in a systematic way starting from a principal diagonal matrix element to find a feasible TSP tour. The matrix reduction, systematic moves in rows and columns, systematic elimination of rows and columns from further consideration, and the “reserved” column declaration, assure that the resulting sequence of nodes and edges forms a complete TSP tour. The process can be repeated from each principal diagonal element. The best TSP tour found can then be used, for example, as an input to another algorithm (e.g. the TABU search, simulated annealing, ant colony optimization, nearest neighbor, or another heuristic) to attempt to improve the tour further. It should be noted, that the proposed technique can also be used for testing of presence of cycles of a proposed solution provided by another algorithm. While the goal of the heuristic algorithm is to attempt to find the optimum tour, optimality cannot be guaranteed.

On the min-cost Traveling Salesman Problem with Drone

Over the past few years, unmanned aerial vehicles (UAV), also known as drones, have been adopted as part of a new logistic method in the commercial sector called “last-mile delivery”. In this novel approach, they are deployed alongside trucks to deliver goods to customers to improve the quality of service and reduce the transportation cost. This approach gives rise to a new variant of the traveling salesman problem (TSP), called TSP with drone (TSP-D). A variant of this problem that aims to minimize the time at which truck and drone finish the service (or, in other words, to maximize the quality of service) was studied in the work of Murray and Chu (2015). In contrast, this paper considers a new variant of TSP-D in which the objective is to minimize operational costs including total transportation cost and one created by waste time a vehicle has to wait for the other. The problem is first formulated mathematically. Then, two algorithms are proposed for the solution. The first algorithm (TSP-LS) was adapted from the approach proposed by Murray and Chu (2015), in which an optimal TSP solution is converted to a feasible TSP-D solution by local searches. The second algorithm, a Greedy Randomized Adaptive Search Procedure (GRASP), is based on a new split procedure that optimally splits any TSP tour into a TSP-D solution. After a TSP-D solution has been generated, it is then improved through local search operators. Numerical results obtained on various instances of both objective functions with different sizes and characteristics are presented. The results show that GRASP outperforms TSP-LS in terms of solution quality under an acceptable running time.

Comments on: Continuous approximation models in freight distribution management

The paper presents a review of the use of continuous approximation modeling for freight distribution management. The authors take a unique approach to reviewing continuous approximation models in the freight distribution literature. They anchor their review around an in-depth analysis of two fundamental results upon which much of the continuous approximation literature is built: the asymptotic approximation formula for the optimal traveling salesman problem (TSP) tour length from Beardwood et al. (1959) and the strip strategy to build implementable TSP tours from Daganzo (1984). Whereas other continuous approximation survey papers focus primarily on application areas and the resulting advances in methodology [e.g., Langevin et al. (1996) for the period up to 1996 and Ansari et al. (2017) for the period from 1996 to 2016], this paper structures the review in the context of these two seminal contributions. As such, the authors create a valuable narrative of the evolution of continuous approximation models.

A Discrete Fruit Fly Optimization Algorithm for the Traveling Salesman Problem.

The fruit fly optimization algorithm (FOA) is a newly developed bio-inspired algorithm. The continuous variant version of FOA has been proven to be a powerful evolutionary approach to determining the optima of a numerical function on a continuous definition domain. In this study, a discrete FOA (DFOA) is developed and applied to the traveling salesman problem (TSP), a common combinatorial problem. In the DFOA, the TSP tour is represented by an ordering of city indices, and the bio-inspired meta-heuristic search processes are executed with two elaborately designed main procedures: the smelling and tasting processes. In the smelling process, an effective crossover operator is used by the fruit fly group to search for the neighbors of the best-known swarm location. During the tasting process, an edge intersection elimination (EXE) operator is designed to improve the neighbors of the non-optimum food location in order to enhance the exploration performance of the DFOA. In addition, benchmark instances from the TSPLIB are classified in order to test the searching ability of the proposed algorithm. Furthermore, the effectiveness of the proposed DFOA is compared to that of other meta-heuristic algorithms. The results indicate that the proposed DFOA can be effectively used to solve TSPs, especially large-scale problems.

<italic>ESync</italic>: Energy Synchronized Mobile Charging in Rechargeable Wireless Sensor Networks

Recent years have witnessed many new promising technologies to power wireless sensor networks, which motivate some fundamental topics to be revisited. Different from energy harvesting, which generates dynamic energy supply, the mobile charger is able to provide a stable and reliable energy supply for sensor nodes and, thus, enables sustainable system operations. While previous mobile charging protocols focus on either the charger travel distance or the charging delay of sensor nodes, in this work, we propose a novel energy synchronized mobile charging (ESync) protocol, which simultaneously reduces both of them. Observing the limitation of traveling salesman problem (TSP)-based solutions, when the nodes' energy consumption is diverse, we construct a set of nested TSP tours based on their energy consumption, and only nodes with low remaining energy are involved in each charging round. Furthermore, we propose the concept of energy synchronization to synchronize the charging request sequence of nodes with their sequence on the TSP tours. Experimentation and simulation demonstrate that ESync can reduce charger travel distance and nodes' charging delay by about 30% and 40%, respectively.

The Traveling Salesman Problem for Lines, Balls, and Planes

We revisit the traveling salesman problem with neighborhoods (TSPN) and propose several new approximation algorithms. These constitute either first approximations (for hyperplanes, lines, and balls in R d , for d ⩾ 3) or improvements over previous approximations achievable in comparable times (for unit disks in the plane). (I) Given a set of n hyperplanes in R d , a traveling salesman problem (TSP) tour whose length is at most O (1) times the optimal can be computed in O ( n ) time when d is constant. (II) Given a set of n lines in R d , a TSP tour whose length is at most O (log 3 n ) times the optimal can be computed in polynomial time for all d . (III) Given a set of n unit balls in R d , a TSP tour whose length is at most O (1) times the optimal can be computed in polynomial time when d is constant.

A k-level data structure for large-scale traveling salesman problems

The problem of data representation is fundamental to the efficiency of search algorithms for the traveling salesman problem (TSP). The computational effort required to perform such tour operations as traversal and subpath reversal considerably influence algorithm design and performance. We propose a new data structure—the k-level satellite tree—for representing a TSP tour with a discussion of properties in the framework of general tour operations. The k-level satellite tree representation is shown to be significantly more efficient than its predecessors for large-scale instances.

Solving Large-Scale TSP Using a Fast Wedging Insertion Partitioning Approach

A new partitioning method, called Wedging Insertion, is proposed for solving large-scale symmetric Traveling Salesman Problem (TSP). The idea of our proposed algorithm is to cut a TSP tour into four segments by nodes’ coordinate (not by rectangle, such as Strip, FRP, and Karp). Each node is located in one of their segments, which excludes four particular nodes, and each segment does not twist with other segments. After the partitioning process, this algorithm utilizes traditional construction method, that is, the insertion method, for each segment to improve the quality of tour, and then connects the starting node and the ending node of each segment to obtain the complete tour. In order to test the performance of our proposed algorithm, we conduct the experiments on various TSPLIB instances. The experimental results show that our proposed algorithm in this paper is more efficient for solving large-scale TSPs. Specifically, our approach is able to obviously reduce the time complexity for running the algorithm; meanwhile, it will lose only about 10% of the algorithm’s performance.

A Gaussian Process Regression Model for the Traveling Salesman Problem

Problem statement: Traveling Salesman Problem (TSP) is a famous NP hard problem. Many approaches have been proposed up to date for solving TSP. We consider a TSP tour as a dependent variable and its corresponding distance a s an independent variable. If a predictive functio n can be formulated from arbitrary sample tours, the optimal tour may be predicted from this function. Approach: In this study, a combined procedure of the Nearest Neighbor (NN) method, Gaussian Process Regression (GPR) and the iterated local sea rch is proposed to solve a deterministic symmetric TSP with a single salesman. The first tour in the s ample is constructed by the nearest neighbor algorithm and it is used to construct other tours b y the random 2-exchange swap. These tours and their total distances are training data for a Gaussian pr ocess regression model. A GPR solution is further improved with the iterated 2-opt method. In the nu merical experiments, our algorithm is tested on many TSP instances and it is compared with the Genetic Algorithm (GA) and the Simulated Annealing (SA) algorithm. Results: The proposed method can find good TSP tours within a reasonable computational time for a wide range of TSP test pro blems. In some cases, it outperforms GA and SA. Conclusion: Our proposed algorithm is promising for solving th e TSP.

On the asymptotic behavior of subtour-patching heuristics in solving the TSP on permuted Monge matrices

We examine the performance of different subtour-patching heuristics for solving the strongly $\mathcal{NP}$ -hard traveling salesman problem (TSP) on permuted Monge matrices. We prove that a well-known heuristic is asymptotically optimal for the TSP on product matrices and k-root cost matrices. We also show that the heuristic is provably asymptotically optimal for general permuted Monge matrices under some mild conditions. Our theoretical results are strongly supported by the findings of a large-scale experimental study on randomly generated numerical examples, which show that the heuristic is not only asymptotically optimal, but also finds optimal TSP tours with high probability that increases with the problem size. Thus the heuristic represents a practical tool to solve large instances of the problem.

Coverage of a Planar Point Set With Multiple Robots Subject to Geometric Constraints

This paper focuses on the assignment of N discrete points among K geometrically constrained robots and determination of the order in which the points should be processed by the robots. This path planning problem is directly motivated by an industrial laser drilling system with two robots that are constrained to translate along a common line while satisfying collision avoidance constraints. The points lie on a planar base plate that translates normal to the axis of motion of the robots. The geometric constraints on the motions of the robots lead to constraints on points that can be processed simultaneously.We use a two step approach to solve the path planning problem: (1) Splitting Problem: Assign the points to the K robots, subject to geometric constraints, to maximize parallel processing of the points. (2) Ordering Problem: Find an order of processing the split points by formulating and solving a multidimensional Traveling Salesman Problem (TSP) in the if-tuple space with an appropriately defined metric to minimize the total travel cost. For K = 2, we solve the splitting problem optimally in O(N3) time by converting it to a maximum cardinality matching problem. Since this is too slow for large datasets, we also provide a greedy O(N log N) algorithm. We provide computational results showing that the greedy algorithm solution is very close to the optimal solution for large datasets. For the ordering problem we present local search based heuristics to improve the multidimensional TSP tour. We give computational results for the ordering problem and for the overall performance gain obtained (over a single robot system) by using our algorithm. Finally, we extend our approach to a K-robot system and give computational results for K = 4.

Fast minimum-weight double-tree shortcutting for metric TSP

The Metric Traveling Salesman Problem (TSP) is a classical NP-hard optimization problem. The double-tree shortcutting method for Metric TSP yields an exponentially-sized space of TSP tours, each of which approximates the optimal solution within, at most, a factor of 2. We consider the problem of finding among these tours the one that gives the closest approximation, that is, the minimum-weight double-tree shortcutting . Burkard et al. gave an algorithm for this problem, running in time O ( n 3 + 2 d n 2 ) and memory O (2 d n 2 ), where d is the maximum node degree in the rooted minimum spanning tree. We give an improved algorithm for the case of small d (including planar Euclidean TSP, where d ≤ 4), running in time O (4 d n 2 ) and memory O (4 d n ). This improvement allows one to solve the problem on much larger instances than previously attempted. Our computational experiments suggest that in terms of the time-quality trade-off, the minimum-weight double-tree shortcutting method provides one of the best existing tour-constructing heuristics.

Min-weight double-tree shortcutting for Metric TSP: Bounding the approximation ratio

The Metric Traveling Salesman Problem (TSP) is a classical NP-hard optimization problem. The double-tree shortcutting method for Metric TSP yields an exponentially-sized space of TSP tours, each of which approximates the optimal solution within at most a factor of 2. We consider the problem of finding among these tours the one that gives the closest approximation, i.e. the minimum-weight double-tree shortcutting. Previously, we gave an efficient algorithm for this problem, and carried out its experimental analysis. In this paper, we address the related question of the worst-case approximation ratio for the minimum-weight double-tree shortcutting method. In particular, we give lower bounds on the approximation ratio in some specific metric spaces: the ratio of 2 in the discrete shortest path metric, 1.622 in the planar Euclidean metric, and 1.666 in the planar Minkowski metric. The first of these lower bounds is tight; we conjecture that the other two bounds are also tight, and in particular that the minimum-weight double-tree method provides a 1.622-approximation for planar Euclidean TSP.

A New Three-Level Tree Data Structure for Representing TSP Tours in the Lin-Kernighan Heuristic

Lin-Kernighan (LK) is the most powerful local search for the Traveling Salesman Problem (TSP). The choice of data structure for tour representation plays a vital role in LK's performance. Binary trees are asymptotically the best tour representation but they perform empirically best only for TSPs with one million or more cities due to a large overhead. Arrays and two-level trees are used for smaller TSPs. This paper proposes a new three-level tree data structure for tour representation. Although this structure is asymptotically not better than the binary tree structure, it performs empirically better than the conventional structures for TSPs having from a thousand to three million cities.