Solving Shortest Path Problems Research Articles

Estimation of Travel Cost between Geographic Coordinates Using Artificial Neural Network: Potential Application in Vehicle Routing Problems

The vehicle routing problem (VRP) attempts to find optimal (minimum length) routes for a set of vehicles visiting a set of locations. Solving a VRP calls for a cost matrix between locations. The size of the matrix grows quadratically with an increasing number of locations, restricting large-sized VRPs from being solved in a reasonable amount of time. The time needed to obtain a cost matrix is expensive when routing engines are used, which solve shortest path problems in the back end. In fact, details on the shortest path are redundant; only distance or time values are necessary for VRPs. In this study, an artificial neural network (ANN) that receives two geo-coordinates as input and provides estimated cost (distance and time) as output is trained. The trained ANN model was able to estimate with a mean absolute percentage error of 7.68%, surpassing the quality of 13.2% with a simple regression model on Euclidean distance. The possibility of using a trained model in VRPs is examined with different implementation scenarios. The experimental results with VRPs confirm that using ANN estimation instead of Euclidean distance produces a better solution, which is verified to be statistically significant. The results also suggest that an ANN can be a better choice than routing engines when the trade-off between response time and solution quality is considered.

Solving shortest path problems using an ant colony algorithm with triangular neutrosophic arc weights

Indeed, one of the most well-known topics in the area of graph theory is the shortest path (SP) problem, which has practical applications in various areas of research, including transportation, communication via networks, life-saving services, fire department services, etc. The edges of the connected SP problems are typically characterized by various numbers in practical applications. In this research paper, we calculate the shortest path using an ant colony optimization (ACO) algorithm with single value triangular neutrosophic numbers as arc weights. The method is used to estimate the shortest path of a neutrosophic network. One numerical example is used to test the suggested method, and outcomes are provided.

On the Convergence of an Efficient and Robust Dynamic Neural Network Concept with Application to Solving Traveling Salesman Problems

In our previous contributions [20, 21, 22], we have clearly demonstrated that the dynamic neural network concept (DNN-concept) for solving shortest path problems (SPP) and traveling salesman problems (TSP) outperforms the best heuristic methods proposed by the literature. However, in our numerous contributions and also according to the literature, the effects of the step sizes of both “decision neurons” and “multiplier neurons” on the convergence properties of the “DNN-concept” are still not investigated. The aim of our contribution is to enrich the literature by investigating, for the first time, the convergence properties of the DNN-concept for solving traveling salesman problems. We develop a mathematical model for the efficient and robust solving the traveling salesman problem (TSP). Based on the numerical study, the convergence properties of the model developed (i.e., the DNN-concept for solving TSP) is investigated. Ranges (or windows) of variation of the parameters of the developed mathematical model are determined (identified) to ensure (guarantee) the detection of the exact TSP solution/tour. In order to validate the mathematical model developed for solving TSP, a bifurcation analysis is carried out using the developed mathematical model. Various bifurcation diagrams are obtained numerically. The bifurcation diagrams obtained reveal the ranges of variation of some key parameters of the model developed to ensure (or guarantee) the convergence of the DNN-concept to the exact TSP-solution (i.e., global minimum). Concrete examples of graphs are considered and various numerical simulations are performed as proof of concept. Finally, a comparison of the results obtained with the results published in [17]-[18] lead to a very good agreement.

Enhancement of Dijkstra Algorithm for Finding Optimal Path

The Dijkstra algorithm is a common method used when solving shortest path problems. It is a graph-based method that compares node distances, selects the shortest subsequent node, and generates an ideal path. However, it is seen that the method struggles with memory utilization, particularly when working with substantial amounts of data or graphs. The purpose of this research work is to improve the algorithm to tackle the current problem. The enhancement is accomplished by providing an approach in which the two closest nodes are combined in each iteration. With this, the conclusion of the study is that the enhanced Dijkstra algorithm, with the application of node combination, was able to reduce the memory usage in comparison to the existing Dijkstra algorithm.

A compact MLCP-based projection recurrent neural network model to solve shortest path problem

ABSTRACT We develop a projection recurrent neural network (RNN) to obtain the solution of the shortest path problem (SPP). Our focus on the paper is to give a compact single-layer structure RNN model to solve the SPP. To present the RNN model, we utilise a mixed linear complementarity problem (MLCP). Moreover, the developed RNN is proved to be globally stable. Finally, some numerical simulations are stated to show the performance of the presented approach. We compare the results with some other methods.

A Comprehensive Comparison of Label Setting Algorithm and Dynamic Programming Algorithm in Solving Shortest Path Problems

In this paper, Label Setting Algorithm and Dynamic Programming Algorithm had been critically examined in determining the shortest path from one source to a destination. Shortest path problems are for finding a path with minimum cost from one or more origin (s) to one or more destination(s) through a connected network. A network of ten (10) cities (nodes) was employed as a numerical example to compare the performance of the two algorithms. Both algorithms arrived at the optimal distance of 11 km, which corresponds to the paths 1→4→5→8→10 ,1→3→5→8→10 , 1→2→6→9→10  and  1→4→6→9→10 . Thus, the problem has multiple shortest paths. The computational results evince the outperformance of Dynamic Programming Algorithm, in terms of time efficiency, over the Label Setting Algorithm. Therefore, to save time, it is recommended to apply Dynamic Programming Algorithm to shortest paths and other applicable problems over the Label-Setting Algorithm.

Using Optimization Algorithms For Solving Shortest Path Problems

Using Optimization Algorithms For Solving Shortest Path Problems

Bellman\u2013Ford algorithm for solving shortest path problem of a network under picture fuzzy environment

An elongation of the novel intuitionistic fuzzy set is a picture fuzzy set theory. The demonstration of this has been used to deal with the abstinence criteria in a decision-making problem. The uncertainty in nature occurs sometimes in real-world problems and amidst them, the prominent one is the shortest path problem (SPP) solving. In the last few years, one of the best algorithms on the network for finding SPP is Bellman–Ford. Due to uncertainty in the decision-making process, it becomes difficult for decision-makers for communicating their point of view and judgment with certainty. We conceive of SPP in this contribution via Bellman's algorithm (BA) for a network with trapezoidal picture fuzzy numbers (TPFNs). We introduce a new algorithm to stand the shortest picture fuzzy path between each pair of nodes. A TPFN is considered for the length of all edges. A numerical example for the validation of the presented algorithm has also been proposed. There has also been relative research with existing techniques showing the benefits of the new algorithm.

On the shortest $$\alpha$$-reliable path problem

In this variant of the constrained shortest path problem, the time of traversing an arc is given by a non-negative continuous random variable. The problem is to find a minimum cost path from an origin to a destination, ensuring that the probability of reaching the destination within a time limit meets a certain reliability threshold. To solve this problem, we extend the pulse algorithm, a solution framework for shortest path problems with side constraints. To allow arbitrary non-negative continuous travel-time distributions, we model the random variables of the travel times using Phase-type distributions and Monte Carlo simulation. We conducted a set of experiments over small- and medium-size stochastic transportation networks with and without spatially-correlated travel times. As an alternative to handling correlations, we present a scenario-based approach in which the distributions of the arc travel times are conditioned to a given scenario (e.g., variable weather conditions). Our methodology and experiments highlight the relevance of considering on-time arrival probabilities and correlations when solving shortest path problems over stochastic transportation networks.

Solving Shortest Path Problems Using Mathematical Literacy Skill Figured Out By Pre-Service Teachers

Mathematical literacy is an important ability to be taught to the mathematics pre-service teacher. The students who have good mathematical literacy skills will be sensitive to the mathematical concepts which are relevant to the phenomenon or problem. One of the problems that is given to mathematics pre-service teacher is the shortest path problem in Graph Theory courses. The purpose of this study is to explore the mathematical literacy abilities of the pre-service teacher in solving the shortest path problems. The research subjects were 56 mathematics pre-service teacher of Universitas PGRI Madiun, Indonesia. The data were collected through problem-solving tests which were then analyzed based on indicators of mathematical literacy ability namely formulate, employ, and interpret. The results shows that the mathematical literacy abilities of pre-service teacher in solving the shortest path problem consist of three categories, namely complete, incomplete, and incomplete mathematical literacy abilities. There are 9 students with complete mathematical literacy skills can formulate, employ, and interpret. There are 18 students with incomplete mathematical literacy skills can formulate and employ. There are 29 students with incomplete mathematical literacy can only formulate.

Solving RFID mobile reader path problem with optimization algorithms

Radio Frequency Identification (RFID) is a one of the fastest growing and most beneficial technologies being adopted by businesses today. One of the important issues is localization of items in a warehouse or business premise and to keep track of the said items, it requires devices which are costly to deploy. This is because many readers need to be placed in a search space. In detecting an object, a reader will only report the signal strength of the tag detected. Once the signal strength report is obtained, the system will compute the coordinates of the RFID tags based on each data grouping. In this paper, algorithms using genetic algorithm, particle swarm, ant colony optimization are proposed to achieve the shortest path for an RFID mobile reader, while covering full search area. In comparison, for path optimization, the mobile reader traverses from one node to the next, moving around encountered obstacles in its path. The tag reading process is iterative, in which the reader arrives at its start point at the end of each round. Based on the shortest path, an algorithm that computes the location of items in the search area is used. The simulation results show that the ACO method works more effectively and efficiently compare to others when solving shortest path problems.

APPLYING DIJKSTRA ALGORITHM FOR SOLVING NEUTROSOPHIC SHORTEST PATH PROBLEM

The selection of shortest path problem is one the classic problems in graph theory. In literature, many algorithms have been developed to provide a solution for shortest path problem in a network. One of common algorithms in solving shortest path problem is Dijkstra’s algorithm. In this paper, Dijkstra’s algorithm has been redesigned to handle the case in which most of parameters of a network are uncertain and given in terms of neutrosophic numbers. Finally, a numerical example is given to explain the proposed algorithm

Convergence results for continuous-time dynamics arising in ant colony optimization

This paper studies the asymptotic behavior of several continuous-time dynamical systems which are analogs of ant colony optimization algorithms that solve shortest path problems. Local asymptotic stability of the equilibrium corresponding to the shortest path is shown under mild assumptions. A complete study is given for a recently proposed model called EigenAnt: global asymptotic stability is shown, and the speed of convergence is calculated explicitly and shown to be proportional to the difference between the reciprocals of the second shortest and the shortest paths.

A Virtual Network Embedding Algorithm for Providing Stronger Connectivity in the Residual Networks

How to efficiently map the nodes and links in a given virtual network (VN) to those in the substrate network (SN) so that the residual substrate network (RSN) can host as many VN requests as possible is a major challenge in virtual network embedding. Most research has developed heuristic algorithms with interactive or two-stage methods. These methods, however, could cause the RSN fragmented into several disconnected components that are insufficient to host a large number of given VN requests. Without loss of generality, we assume that SNs, as the Internet, are small world and scale free, meaning that the average number of hops between any two nodes is a small constant and that most nodes have small degrees while only a small number of nodes have large degrees. Taking advantage of these two properties, we devise in this paper a new twostage VN embedding approach to improve the connectivity of the residual substrate network so that it has the capacity to host more VN requests. Our algorithm uses a greedy strategy that maps neighboring VN nodes to substrate nodes whose distances are bounded by a small constant. We also map the edge nodes of given VN, i.e., nodes with small degrees, to nodes with large degrees in the SN. We then map the links by solving shortest path problems. Our experimental evaluations show that our algorithms offer better mappings and results in significantly fewer rejections for VN requests.

Distant Location Selection Using Genetic Algorithm for Live Migration Method in OpenFlow Networks

In the last decade, the massive undersea earthquake and Tsunami occurred in Taiwan on Tuesday December 26, 2006 and Japan on Friday March 11, 2011. Those natural disasters had affected the telecommunication in the worldwide. They disrupted the infrastructures including not only Internet services but also business and financial transactions. Thus, keeping the system and functions alive are particularly crucial to many organizations relying on them. Migration is one of the solutions to keep the systems alive. This paper introduces a migration technique to migrate network systems from origin sites to other remote sites. We propose a Genetic Algorithm (GA) approach to solve shortest path problems for selecting the best possible remote site prior to initiate a migration. Network virtualization in OpenFlow technology is particularly valuable in the implementation to relocate the systems by using network segmentation technique.

Solving shortest path problems with a weight constraint and replenishment arcs

This paper tackles a generalization of the weight constrained shortest path problem (WCSPP) in a directed network with replenishment arcs that reset the accumulated weight along the path to zero. Such situations arise, for example, in airline crew pairing applications, where the weight represents duty hours, and replenishment arcs represent crew overnight rests; and also in aircraft routing, where the weight represents time elapsed, or flight time, and replenishment arcs represent maintenance events. In this paper, we review the weight constrained shortest path problem with replenishment (WCSPP-R), develop preprocessing methods, extend existing WCSPP algorithms, and present new algorithms that exploit the inter-replenishment path structure. We present the results of computational experiments investigating the benefits of preprocessing and comparing several variants of each algorithm, on both randomly generated data, and data derived from airline crew scheduling applications.

On the Stable Paths Problem

The Border Gateway Protocol (BGP) is the interdomain routing protocol used to exchange routing information between Autonomous Systems (ASes) in the internet today. While intradomain routing protocols such as RIP are basically distributed algorithms for solving shortest path problems, the graph theoretic problem that BGP is trying to solve is the stable paths problem (SPP). Unfortunately, unlike shortest path problems, it has been shown that instances of SPP can fail to have a solution, and so BGP can fail to converge. We define a fractional version of SPP and show that all instances of fractional SPP have solutions. We also show that there are polynomial time reductions from a number of well-known graph problems to SPP. For example, finding stable matchings in hypergraphic preference systems (a generalization of graph stable matchings to the case of hypergraphs) and computing kernels in directed graphs are both polynomial time reducible to SPP. These reductions remain valid in the fractional case. Thus the existence of a polynomial time algorithm for computing fractional solutions to SPP would imply polynomial time algorithms for fractional solutions to these other problems as well.

ON SOLVING SHORTEST PATHS WITH A LEAST-SQUARES PRIMAL-DUAL ALGORITHM

Recently a new least-squares primal-dual (LSPD) algorithm, that is impervious to degeneracy, has effectively been applied to solving linear programming problems by Barnes et al., 2002. In this paper, we show an application of LSPD to shortest path problems with nonnegative arc length is equivalent to the Dijkstra's algorithm. We also compare the LSPD algorithm with the conventional primal-dual algorithm in solving shortest path problems and show their difference due to degeneracy in solving the 1-1 shortest path problems.

Solving shortest path problem using particle swarm optimization

This paper presents the investigations on the application of particle swarm optimization (PSO) to solve shortest path (SP) routing problems. A modified priority-based encoding incorporating a heuristic operator for reducing the possibility of loop-formation in the path construction process is proposed for particle representation in PSO. Simulation experiments have been carried out on different network topologies for networks consisting of 15–70 nodes. It is noted that the proposed PSO-based approach can find the optimal path with good success rates and also can find closer sub-optimal paths with high certainty for all the tested networks. It is observed that the performance of the proposed algorithm surpasses those of recently reported genetic algorithm based approaches for this problem.

Approximating Fractional Packings and Coverings in O(1/epsilon) Iterations

We adapt a method proposed by Nesterov [Math. Program. Ser. A, 103 (2005), pp. 127-152] to design an algorithm that computes $\epsilon$-optimal solutions to fractional packing problems by solving $O(\epsilon^{-1}\sqrt{Kn\ln(m)})$ separable convex quadratic programs, where n is the number of variables, m is the number of constraints, and K is the maximum number of nonzero elements in any constraint. We show that the quadratic program can be approximated to any degree of accuracy by an appropriately defined piecewise-linear program. For the special case of the maximum concurrent flow problem on a graph $G = (V,E)$ with rational capacities and demands, we obtain an algorithm that computes an $\epsilon$-optimal flow by solving shortest path problems, i.e., problems in which the number of shortest paths computed grows as $O(\epsilon^{-1} \log(\epsilon^{-1}))$ in $\epsilon$ and polynomially in the size of the problem. In contrast, previous algorithms required $\Omega(\epsilon^{-2})$ iterations. We also describe extensions to the maximum multicommodity flow problem, the pure covering problem, and mixed packing-covering problem.