Euclidean Traveling Salesperson Problem Research Articles

Geometric reasoning on the euclidean traveling salesperson problem in answer set programming1

The Traveling Salesperson Problem (TSP) is one of the best-known problems in computer science. Many instances and real world applications fall into the Euclidean TSP special case, in which each node is identified by its coordinates on the plane and the Euclidean distance is used as cost function. It is worth noting that in the Euclidean TSP more information is available than in the general case; in a previous publication, the use of geometric information has been exploited to speedup TSP solving for Constraint Logic Programming (CLP) solvers. In this work, we study the applicability of geometric reasoning to the Euclidean TSP in the context of an ASP computation. We compare experimentally a classical ASP approach to the TSP and the effect of the reasoning based on geometric properties. We also compare the speedup of the additional filtering based on geometric information on an ASP solver and a CLP on Finite Domain (CLP(FD)) solver.

Traveling Salesperson Problem with Simple Obstacles: The Role of Multidimensional Scaling and the Role of Clustering

The objective of the traveling salesperson problem (TSP) is to find a shortest tour through all nodes in a graph. Euclidean TSPs are traditionally represented with “cities” placed on a 2D plane. When straight line obstacles are added to the plane, a tour has to visit all cities while going around obstacles. The resulting problem with obstacles remains metric, but is not Euclidean because the shortest paths are no longer straight lines. We first revise a previous version of multiresolution graph pyramid by modifying the hierarchical clustering stage. Next, we describe two new experiments with human subjects. In the first experiment, the effect of the length of obstacles on the quality of tours produced by subjects was tested with three problem sizes. Long obstacles affect the tours to a greater degree than short obstacles, but long obstacles create obvious clusters and limit the ways in which the tours can be produced. In the second experiment we evaluated the degree to which Multidimensional Scaling (MDS) can compensate for the presence of obstacles. The results show that although MDS approximation can compensate to a large degree for the presence of obstacles, it cannot fully account for human performance. This fact suggests that mental representation of a TSP with obstacles is not Euclidean. Instead, it is likely to be based on hierarchical clustering in which pairwise distances represent the shortest paths around obstacles.

Perceived optimality of competing solutions to the Euclidean travelling salesperson problem

The Travelling Salesperson Problem (TSP) is a nondeterministic-polynomial hard (NP-hard) combinatorial problem that occurs in a wide range of industrial domains, including logistics, route finding, and computer wiring. Interestingly, despite the problem’s inherent computational difficulty, when presented in Euclidean space (ETSP), human participants can produce close-to-optimal solutions in near-linear time. However, when asked to compare and select the most optimum solution from a set of pre-defined competing solution options, participants can struggle. In this study we investigate this paradox by asking participants to compare four closed-loop Euclidean TSP solutions, in order to determine which solution they perceived to have the most optimal tour cost. We hypothesise that the extracted geometric properties have an effect on stimulus selection in a discrimination task (selection or no selection). Accordingly, we extracted four geometric properties from competing stimuli in order to create a perceptual activation function. Predictive analytics demonstrated that a classification model could identify the most optimal solution 97% of the time using the perceptual activation scores alone, yet human participants only correctly determined the most optimal solution 47% of the time. Mixed-effects models suggest that ‘likelihood of stimulus selection’ can be modelled as a function of the weighted coefficients of competing perceptual activation scores within each trial; however only a small amount of the variance is explained by these perceptual activation scores. Finally, a drift–diffusion model was used to create a theoretical framework of how likelihood of stimulus selection is influenced by competing perceptual activators. Our study highlights a novel way of extracting and analysing the importance of geometric properties that influence ETSP discrimination tasks, and links this analysis to human behaviour when discriminating between competing ETSP solutions.

A Parameterized Runtime Analysis of Evolutionary Algorithms for the Euclidean Traveling Salesperson Problem

We contribute to the theoretical understanding of evolutionary algorithms and carry out a parameterized analysis of evolutionary algorithms for the Euclidean traveling salesperson problem (Euclidean TSP). We exploit structural properties related to the optimization process of evolutionary algorithms for this problem and use them to bound the runtime of evolutionary algorithms. Our analysis studies the runtime in dependence of the number of inner points $k$ and shows that simple evolutionary algorithms solve the Euclidean TSP in expected time O(nk(2k-1)!). Moreover, we show that, under reasonable geometric constraints, a locally optimal 2-opt tour can be found by randomized local search in expected time $O(n2kk!).

Constraint Programming Algorithms for Route Planning Exploiting Geometrical Information

Problems affecting the transport of people or goods are plentiful in industry and commerce and they also appear to be at the origin of much more complex problems. In recent years, the logistics and transport sector keeps growing supported by technological progress, i.e. companies to be competitive are resorting to innovative technologies aimed at efficiency and effectiveness. This is why companies are increasingly using technologies such as Artificial Intelligence (AI), Blockchain and Internet of Things (IoT). Artificial intelligence, in particular, is often used to solve optimization problems in order to provide users with the most efficient ways to exploit available resources. In this work we present an overview of our current research activities concerning the development of new algorithms, based on CLP techniques, for route planning problems exploiting the geometric information intrinsically present in many of them or in some of their variants. The research so far has focused in particular on the Euclidean Traveling Salesperson Problem (Euclidean TSP) with the aim to exploit the results obtained also to other problems of the same category, such as the Euclidean Vehicle Routing Problem (Euclidean VRP), in the future.

Improved Filtering for the Euclidean Traveling Salesperson Problem in CLP(FD)

The Traveling Salesperson Problem (TSP) is one of the best-known problems in computer science. The Euclidean TSP is a special case in which each node is identified by its coordinates on the plane and the Euclidean distance is used as cost function.
 Many works in the Constraint Programming (CP) literature addressed the TSP, and use as benchmark Euclidean instances; however the usual approach is to build a distance matrix from the points coordinates, and then address the problem as a TSP, disregarding the information carried by the points coordinates for constraint propagation.
 In this work, we propose to use geometric information, present in Euclidean TSP instances, to improve the filtering power. In order to have a declarative approach, we implemented the filtering algorithms in Constraint Logic Programming on Finite Domains (CLP(FD)).

Replica symmetry and replica symmetry breaking for the traveling salesperson problem.

We study the energy landscape of the traveling salesperson problem (TSP) using exact ground states and a novel linear programming approach to generate excited states with closely defined properties. We look at four different ensembles, notably the classic finite dimensional Euclidean TSP and the mean-field-like (1,2)-TSP, which has its origin directly in the mapping of the Hamiltonian circuit problem on the TSP. Our data supports previous conjectures that the Euclidean TSP does not show signatures of replica symmetry breaking neither in two nor in higher dimension. On the other hand the (1,2)-TSP exhibits some signature which does not exclude broken replica symmetry, making it a candidate for further studies in the future.

Human behaviour in the Euclidean Travelling Salesperson Problem: Computational modelling of heuristics and figural effects

The Travelling Salesperson Problem (TSP) describes a situation where an imaginary individual wishes to visit multiple cities once before returning to his/her own city. This type of problem is known as a nondeterministic polynomial (NP) hard problem, since the factorial number of solutions results in it being impractical to solve using exhaustive processing. Interestingly, when presented as a Euclidean graph (i.e., ETSP), humans identify near optimal solutions almost effortlessly, despite billions of possible tours. In this study, we consider human processing of the ETSP, and introduce the reader to a number of factors that literature proposes as impacting human performance. We hypothesise that: (i) human ETSP activity may be modelled by considering the quotient relationship between node-to-node and node-to-centroid distances; and (ii) consideration of figural effects can optimise automated TSP solution generation. In this paper human processing based heuristics are developed, i.e. replacing the cost function within the nearest neighbour algorithm, to guide node selection. Results showed that the quotient relationship between node-to-node and node-to-centroid distances can be used to closely model average human performance, across a range of ETSP graphs. Interestingly, however, additional consideration of graph figural effects (e.g. distance between boundary points in the convex hull, standard deviation of distances between nodes that make up the convex hull, and number of nodes in the convex hull) results in significantly improved tour costs.

Leveraging TSP Solver Complementarity through Machine Learning.

The Travelling Salesperson Problem (TSP) is one of the best-studied NP-hard problems. Over the years, many different solution approaches and solvers have been developed. For the first time, we directly compare five state-of-the-art inexact solvers-namely, LKH, EAX, restart variants of those, and MAOS-on a large set of well-known benchmark instances and demonstrate complementary performance, in that different instances may be solved most effectively by different algorithms. We leverage this complementarity to build an algorithm selector, which selects the best TSP solver on a per-instance basis and thus achieves significantly improved performance compared to the single best solver, representing an advance in the state of the art in solving the Euclidean TSP. Our in-depth analysis of the selectors provides insight into what drives this performance improvement.

Acknowledging crossing-avoidance heuristic violations when solving the Euclidean travelling salesperson problem.

If a salesperson aims to visit a number of cities only once before returning home, which route should they take to minimise the total distance/cost? This combinatorial optimization problem is called the travelling salesperson problem (TSP) and has a rapid growth in the number of possible solutions as the number of cities increases. Despite its complexity, when cities and routes are represented in 2D Euclidean space (ETSP), humans solve the problem with relative ease, by applying simple visual heuristics. One of the most important heuristics appears to be the avoidance of path crossings, which will always result in more optimal solutions than tours that contain crossings. This study systematically investigates whether the occurrence of crossings is impacted by geometric properties by modelling their relationship using binomial logistic regression as well as random forests. Results show that properties, such as the number of nodes making up the convex hull, the standard deviation of the angles between nodes, the average distance between all nodes in the graph, the total number of nodes in the graph, and the tour cost (i.e., a measure of performance), are significant predictors of whether crossings are likely to occur.

Expected Fitness Gains of Randomized Search Heuristics for the Traveling Salesperson Problem.

Randomized search heuristics are frequently applied to NP-hard combinatorial optimization problems. The runtime analysis of randomized search heuristics has contributed tremendously to our theoretical understanding. Recently, randomized search heuristics have been examined regarding their achievable progress within a fixed-time budget. We follow this approach and present a fixed-budget analysis for an NP-hard combinatorial optimization problem. We consider the well-known Traveling Salesperson Problem (TSP) and analyze the fitness increase that randomized search heuristics are able to achieve within a given fixed-time budget. In particular, we analyze Manhattan and Euclidean TSP instances and Randomized Local Search (RLS), (1+1)EA and (1+[Formula: see text])EA algorithms for the TSP in a smoothed complexity setting, and derive the lower bounds of the expected fitness gain for a specified number of generations.

Parameterized Runtime Analyses of Evolutionary Algorithms for the Planar Euclidean Traveling Salesperson Problem

Parameterized runtime analysis seeks to understand the influence of problem structure on algorithmic runtime. In this paper, we contribute to the theoretical understanding of evolutionary algorithms and carry out a parameterized analysis of evolutionary algorithms for the Euclidean traveling salesperson problem (Euclidean TSP). We investigate the structural properties in TSP instances that influence the optimization process of evolutionary algorithms and use this information to bound their runtime. We analyze the runtime in dependence of the number of inner points k. In the first part of the paper, we study a [Formula: see text] EA in a strictly black box setting and show that it can solve the Euclidean TSP in expected time [Formula: see text] where A is a function of the minimum angle [Formula: see text] between any three points. Based on insights provided by the analysis, we improve this upper bound by introducing a mixed mutation strategy that incorporates both 2-opt moves and permutation jumps. This strategy improves the upper bound to [Formula: see text]. In the second part of the paper, we use the information gained in the analysis to incorporate domain knowledge to design two fixed-parameter tractable (FPT) evolutionary algorithms for the planar Euclidean TSP. We first develop a [Formula: see text] EA based on an analysis by M. Theile, 2009, "Exact solutions to the traveling salesperson problem by a population-based evolutionary algorithm," Lecture notes in computer science, Vol. 5482 (pp. 145-155), that solves the TSP with k inner points in [Formula: see text] generations with probability [Formula: see text]. We then design a [Formula: see text] EA that incorporates a dynamic programming step into the fitness evaluation. We prove that a variant of this evolutionary algorithm using 2-opt mutation solves the problem after [Formula: see text] steps in expectation with a cost of [Formula: see text] for each fitness evaluation.

On Evaluating Human Problem Solving of Computationally Hard Problems

This article is concerned with how computer science, and more exactly computational complexity theory, can inform cognitive science. In particular, we suggest factors to be taken into account when investigating how people deal with computational hardness. This discussion will address the two upper levels of Marr’s Level Theory: the computational level and the algorithmic level. Our reasons for believing that humans indeed deal with hard cognitive functions are threefold: (1) Several computationally hard functions are suggested in the literature, e.g., in the areas of visual search, visual perception and analogical reasoning, linguistic processing, and decision making. (2) People appear to be attracted to computationally hard recreational puzzles and games. Examples of hard puzzles include Sudoku, Minesweeper, and the 15-Puzzle. (3) A number of research articles in the area of human problem solving suggest that humans are capable of solving hard computational problems, like the Euclidean Traveling Salesperson Problem, quickly and near-optimally. This article gives a brief introduction to some theories and foundations of complexity theory and motivates the use of computationally hard problems in human problem solving with a short survey of known results of human performance, a review of some computationally hard games and puzzles, and the connection between complexity theory and models of cognitive functions. We aim to illuminate the role that computer science, in particular complexity theory, can play in the study of human problem solving. Theoretical computer science can provide a wealth of interesting problems for human study, but it can also help to provide deep insight into these problems. In particular, we discuss the role that computer science can play when choosing computational problems for study and designing experiments to investigate human performance. Finally, we enumerate issues and pitfalls that can arise when choosing computationally hard problems as the subject of study, in turn motivating some interesting potential future lines of study. The pitfalls addressed include: choice of presentation and representation of problem instances, evaluation of problem comprehension, and the role of cognitive support in experiments. Our goal is not to exhaustively list all the ways in which these choices may impact experimental studies, but rather to provide a few simple examples in order to highlight possible pitfalls.

Clustering, Randomness, and Regularity: Spatial Distributions and Human Performance on the Traveling Salesperson Problem and Minimum Spanning Tree Problem

We investigated human performance on the Euclidean Traveling Salesperson Problem (TSP) and Euclidean Minimum Spanning Tree Problem (MST-P) in regards to a factor that has previously received little attention within the literature: the spatial distributions of TSP and MST-P stimuli. First, we describe a method for quantifying the relative degree of clustering, randomness or regularity within point distributions. We then review evidence suggesting this factor might influence human performance on the two problem types. Following this we report an experiment in which the participants were asked to solve TSP and MST-P test stimuli that had been generated to be either highly clustered, random, or highly regular. The results indicate that for both the TSP and MST-P the participants tended to produce better quality solutions when the stimuli were highly clustered compared to random, and similarly, better quality solutions for random compared to highly regular stimuli. It is suggested that these results provide support for the ideas that human solvers attend to salient clusters of nodes when solving these problems, and that a similar process (or series of processes) may underlie human performance on these two tasks.

The Wisdom of the Crowd in Combinatorial Problems

The "wisdom of the crowd" phenomenon refers to the finding that the aggregate of a set of proposed solutions from a group of individuals performs better than the majority of individual solutions. Most often, wisdom of the crowd effects have been investigated for problems that require single numerical estimates. We investigate whether the effect can also be observed for problems where the answer requires the coordination of multiple pieces of information. We focus on combinatorial problems such as the planar Euclidean traveling salesperson problem, minimum spanning tree problem, and a spanning tree memory task. We develop aggregation methods that combine common solution fragments into a global solution and demonstrate that these aggregate solutions outperform the majority of individual solutions. These case studies suggest that the wisdom of the crowd phenomenon might be broadly applicable to problem-solving and decision-making situations that go beyond the estimation of single numbers.

Continuous reformulations and heuristics for the Euclidean travelling salesperson problem

We consider continuous reformulations of the Euclidean travelling salesperson problem (TSP), based on certain clustering problem formulations. These reformulations allow us to apply a generalisation with perturbations of the Weiszfeld algorithm in an attempt to find local approximate solutions to the Euclidean TSP.

Perceptual or Analytical Processing? Evidence from Children's and Adult's Performance on the Euclidean Traveling Salesperson Problem

The Euclidean Traveling Salesperson Problem (E-TSP) is a useful task to study how humans optimize when faced with computational intractability. It has been found that humans are capable of finding high-quality solutions for E-TSP in a relatively short time and with seemingly little cognitive effort. This observation has led to two general proposals: The high quality of performance on E-TSP reflects (a) the output of automatic and innate perceptual processes or (b) a fundamentally intelligent approach to the task by humans. An experiment was performed to compare performance of three age groups (7- year-olds, 12-year-olds, and adults). Our findings provide corroborating evidence that high-level performance on E-TSP is attainable with perceptual processing alone. At the same time, the exceptionally high performance of adult participants suggests that they additionally exploit their cognitive-analytic skills to improve upon what their visual systems provide.

Convex hull and tour crossings in the Euclidean traveling salesperson problem: implications for human performance studies.

Recently there has been growing interest among psychologists in human performance on the Euclidean traveling salesperson problem (E-TSP). A debate has been initiated on what strategy people use in solving visually presented E-TSP instances. The most prominent hypothesis is the convex-hull hypothesis, originally proposed by MacGregor and Ormerod (1996). We argue that, in the literature so far, there is no evidence for this hypothesis. Alternatively we propose and motivate the hypothesis that people aim at avoiding crossings.

On the Stochastic Euclidean Travelling Salesperson Problem for Distributions with Unbounded Support

We study the asymptotic behavior of the shortest tour Tn through n points X1, …, Xn in ℝd (d ≥ 2), where (Xi)i≥1 are i.i.d. random variables, whose density f(x) has an unbounded support. Beardwood et al. [2] conjectured that Tn/n1−1/d converges a.s. if and only if ∫ f(x)1−1/d dx < ∞ and ∫ f(x)‖x‖d/(d−1) dx < ∞. We disprove this conjecture, and we show that the second integrability condition is not strong enough. We give a stronger condition that is optimal (but not necessary).

The searching over separators strategy to solve some NP-hard problems in subexponential time

In this paper we propose a new strategy for designing algorithms, called the searching over separators strategy. Suppose that we have a problem where the divide-and-conquer strategy can not be applied directly. Yet, also suppose that in an optimal solution to this problem, there exists a separator which divides the input points into two parts,A d andC d, in such a way that after solving these two subproblems withA d andC d as inputs, respectively, we can merge the respective subsolutions into an optimal solution. Let us further assume that this problem is an optimization problem. In this case our searching over separators strategy will use a separator generator to generate all possible separators. For each separator, the problem is solved by the divide-and-conquer strategy. If the separator generator is guaranteed to generate the desired separator existing in an optimal solution, our searching over separators strategy will always produce an optimal solution. The performance of our approach will critically depend upon the performance of the separator generator. It will perform well if the total number of separators generated is relatively small. We apply this approach to solve the discrete EuclideanP-median problem (DEPM), the discrete EuclideanP-center problem (DEPC), the EuclideanP-center problem (EPC), and the Euclidean traveling salesperson problem (ETSP). We propose $$O(n^{o(\sqrt P )} )$$ algorithms for the DEPM problem, the DEPC problem, and the EPC problem, and we propose an $$O(n^{o(\sqrt n )} )$$ algorithm for the ETSP problem, wheren is the number of input points.