Shortest Path Problem Research Articles

Improved Floyd-Warshall Algorithm for Solving Travelling Salesman Problem

The shortest path problem is a fundamental challenge in graph theory, focused on identifying the most efficient routes between nodes in a network. It stands as one of the extensively researched combinatorial optimization problems. In this paper, we explained the fundamental concepts of shortest path algorithms, with a particular emphasis on Floyd Warshall algorithm. We apply the algorithm and showed how this algorithm can be effectively employed to determine the shortest path in various scenarios. Furthermore, this paper addresses real-life applications, particularly focusing on the traveling salesman problem. Individuals often have the option to travel along different routes to reach their destination. However, selecting suboptimal routes can result in time-consuming journeys. To tackle this issue, we apply the Floyd Warshall algorithm to solve the traveling salesman problem (TSP). Additionally, we demonstrate the enhancement of the Floyd Warshall algorithm's efficiency for TSP using the rectangular algorithm. This paper concludes with a comparative analysis between the original and improved Floyd Warshall algorithms, emphasizing the computational reduction achieved through the proposed enhancements. Jagannath University Journal of Science, Volume 11, Number 1, June 2024, pp. 45−52

Polynomial Time Algorithm for Shortest Paths in Interval Temporal Graphs

We develop a polynomial time algorithm for the single-source all destinations shortest paths problem for interval temporal graphs (ITGs). While a polynomial time algorithm for this problem is known for contact sequence temporal graphs (CSGs), no such prior algorithm is known for ITGs. We benchmark our ITG algorithm against that for CSGs using datasets that can be solved using either algorithm. Using synthetic datasets, experimentally, we show that our algorithm for ITGs obtains a speedup of up to 32.5 relative to the state-of-the-art algorithm for CSGs.

Recoverable Robust Shortest Path Problem Under Interval Budgeted Uncertainty Representations

ABSTRACTThis article deals with the recoverable robust shortest path problem under interval uncertainty representations. In this problem, a first‐stage path is computed, which can be modified to some extent after observing changes in the cost structure. The uncertain second‐stage arc costs are modeled by intervals, and the robust min–max criterion is used to compute an optimal solution. The problem is known to be strongly NP‐hard and also hard to approximate in general digraphs. However, until now its complexity for acyclic digraphs was unknown. In this article, it is shown that the problem in acyclic digraphs can be solved in polynomial time for the traditional interval uncertainty and all natural neighborhoods known from the literature. More efficient algorithms for layered and arc series‐parallel digraphs are constructed. Hardness results for general digraphs are also strengthened. Finally, some exact and approximate methods of solving the problem under interval budgeted uncertainty are proposed.

A branch-and-price approach for the nurse rostering problem with multiple units

In this paper, we study the nurse rostering problem that considers multiple units and many soft time-related constraints. An efficient branch and price solution approach that relies on a fast algorithm to solve the pricing subproblem of the column generation process is presented. For the nurse rostering problem, its pricing subproblem can be formulated as a shortest path problem with resource constraints, which has been the backbone of several solutions for several classical problems like vehicle routing problems. However, approaches that perform well on these problems cannot be used since most constraints in the nurse rostering problem are soft. Based on ideas borrowed from global constraints in constraint programming to model rostering problems, an efficient dynamic programming algorithm with novel label definitions and dominating rules specific to soft time-related constraints is proposed. In addition, several acceleration strategies are employed to improve the branch and price algorithm. Computational results on two sets of instances ranging from 2 to 4 units over a horizon of 2 or 4 weeks demonstrate the effectiveness of the proposed algorithms. The accelerated dynamic programming algorithm is able to solve pricing subproblems to optimality with the fewest extended labels. When contrasted with Cplex, the branch and price approach yields solutions that are either equivalent or superior across all instances.

Ship model-based route optimisation for decision support in deep sea shipping

Abstract We present a new approach and route optimization methodology to support analysis and planning of vessel and fleet performance in deep sea shipping for green, energy efficient, and safe navigation. The developed methodology combines the ship technical characteristics based on a ship model developed for a particular vessel type and energy-saving technology options and an optimization algorithm taking into account weather conditions. Optimization involves two stages: graph construction and Dynamic Programming labelling algorithm implemented to solve the shortest path problem with variable speed. The new approach is implemented as part of a decision-support tool EcoRouter enabling the user to conduct analysis of safe and Pareto optimal solutions. Several applications to support real fleet planning and ship performance analysis have been identified including project engineering for future energy-saving ship technologies, for example, wind-assisted propulsion.

Solving the Robust Shortest Path Problem with Multimodal Transportation

This paper explores the challenges of finding robust shortest paths in multimodal transportation networks. With the increasing complexity and uncertainties in modern transportation systems, developing efficient and reliable routing strategies that can adapt to various disruptions and modal changes is essential. By incorporating practical constraints in parameter uncertainty, this paper establishes a robust shortest path mixed-integer programming model based on a multimodal transportation network under transportation time uncertainty. To solve robust shortest path problems with multimodal transportation, we propose a modified Dijkstra algorithm that integrates parameter uncertainty with multimodal transportation. The effectiveness of the proposed multimodal transportation shortest path algorithm is verified using empirical experiments on test sets of different scales and a comparison of the runtime using a commercial solver. The experimental results on the multimodal transportation networks demonstrate the effectiveness of our approach in providing robust and efficient routing solutions. The results demonstrate that the proposed method can generate optimal solutions to the robust shortest path problem in multimodal transportation under time uncertainty and has practical significance.

Shortest Path Problem under Pythagorean Fuzzy Network Using Dijkstra’s Algorithm

In this article investigates the Shortest Path Problem (SPP) in uncertain settings using Pythagorean fuzzy numbers (PFNs). A heuristic methodology using modified Dijkstra's algorithm is developed to exploit error, uncertainty, and partial truth for a low-cost solution. The method calculates minimum distance between starting and destination nodes and evaluates its effectiveness. Pythagorean fuzzy graphs can handle ambiguous situations.

The Multi-Objective Shortest Path Problem with Multimodal Transportation for Emergency Logistics

The optimization of emergency logistical transportation is crucial for the timely dispatch of aid and support to affected areas. By incorporating practical constraints into emergency logistics, this study establishes a multi-objective shortest path mixed-integer programming model based on a multimodal transportation network. To solve multi-objective shortest path problems with multimodal transportation, we design an ideal point method and propose a procedure for constructing the complete Pareto frontier based on the k-shortest path multi-objective algorithm. We use modified Dijkstra and Floyd multimodal transportation shortest path algorithms to build a k-shortest path multi-objective algorithm. The effectiveness of the proposed multimodal transportation shortest path algorithm is verified using empirical experiments carried out on test sets of different scales and a comparison of the runtime using a commercial solver. The results show that the modified Dijkstra algorithm has a runtime that is 100 times faster on average than the modified Floyd algorithm, which highlights its greater applicability in large-scale multimodal transportation networks, demonstrating that the proposed method both has practical significance and can generate satisfactory solutions to the multi-objective shortest path problem with multimodal transportation in the context of emergency logistics.

LPulse: An efficient algorithm for service function chain placement and routing with delay guarantee

Modern network services increasingly depend on the effective orchestration of the Service Function Chain (SFC) with stringent end-to-end delay guarantees. To achieve this, the Delay-Constrained Service Function Chain Placement and Routing (DC-SFCPR) problem must be addressed. This problem involves the optimal selection of nodes for placing network functions and routes that adhere to a specific sequence of service functions, to minimize network bandwidth and CPU costs while strictly adhering to stringent end-to-end delay constraints. The DC-SFCPR problem is NP-hard and existing algorithms either fail to guarantee strict delay constraints or are computationally expensive, making them unsuitable for expanding network topologies. We propose the LPulse algorithm, designed to efficiently solve the DC-SFCPR problem. This algorithm utilizes a layered graph to embed the requirements of service functions, transforming the DC-SFCPR problem into a Delay-Constrained Shortest Path (DCSP) problem. The LPulse algorithm then applies Pulse, a depth-first search framework enhanced with efficient pruning strategies, and incorporates two novel acceleration strategies to solve the DCSP problem. We prove that LPulse ensures the optimality of solutions. Evaluations conducted across various topologies, with node scales ranging from 22 to 10,000, show that LPulse surpasses existing algorithms in both solution quality and speed. For instance, the number of cases meeting strict delay constraints with LPulse is 1.9× that of those solved by deep reinforcement learning algorithms; furthermore, its solving efficiency is 4.9× that of the highest-performing existing optimal algorithm, the LagrangianKsp algorithm.

Railway alignment optimization in regions with densely-distributed obstacles based on semantic topological maps

Railway alignment development in a study area with densely-distributed obstacles, in which regions favorable for alignments are isolated (termed an isolated island effect, i.e., IIE), is a computation-intensive and time-consuming task. To enhance search efficiency and solution quality, an environmental suitability analysis is conducted to identify alignment-favorable regions (AFRs), focusing the subsequent alignment search on these areas. Firstly, a density-based clustering algorithm (DBSCAN) and a specific criterion are customized to distinguish AFR distribution patterns: continuously-distributed AFRs, obstructed effects, and IIEs. Secondly, a study area characterized by IIEs is represented with a semantic topological map (STM), integrating between-island and within-island paths. Specifically, between-island paths are derived through a multi-directional scanning strategy, while within-island paths are optimized using a Floyd-Warshall algorithm. To this end, the intricate alignment optimization problem is simplified into a shortest path problem, tackled with conventional shortest path algorithms (of which Dijkstra’s algorithm is adopted in this work). Lastly, the proposed method is applied to a real case in a mountainous region with karst landforms. Numerical results indicate its superior performance in both construction costs and environmental suitability compared to human designers and a prior alignment optimization method.

Optimizing multi-channel procurement planning under disruption risks

Procurement plays a vital role in supply chain management, which directly affects the production, delivery, and reputation of enterprises. This paper investigates the multi-channel procurement planning problem (MCPP-R) considering the impact of disruption risk, uncertain demand and spot-market price. To meet the customer demand, the buyer can make purchases by signing a long-term contract with the primary supplier (PS), which offers certain advantages in price but may be susceptible to disruption risks. Alternatively, the buyer can enter into an option contract to retain the right to purchase from a backup supplier (BS). Additionally, the buyer can also purchase from the spot market with uncertain prices. During the procurement process, the buyer needs to decide the quantity to order from the PS and the quantity to reserve from the BS first, and then decide the quantities to reserve from a BS and the spot market respectively when the customer demand realization and spot-market price emerge. The MCPP-R is to find the optimal procurement portfolio solution with the minimum expected total procurement cost, formulated as a two-stage stochastic programming model. Due to the non-convex and non-continuous nature, the MCPP-R model is solved optimally by transforming equivalently into a shortest-path problem (SPP) with constraints. The solution method does not have specific requirements for the distribution of uncertain demand and spot-market price. We conduct extensive numerical experiments to analyze the impact of risk and uncertainty in demand and spot-market price on the procurement plan. The experimental analysis indicates that BS plays a crucial role in most cases, particularly with high disruption probability or high spot-market price.

An Efficient Approach for Shortest Path Planning in Automotive Navigation System

Emerging automotive engineering solutions extensively rely on navigation tools. Shortest path evaluation is core competency of optimal path planning, one of the most critical component of any navigation application. Fibonacci heap variant of state-of-the-art Dijkstra’s algorithm solve single source shortest path problems in asymptotic O(E + V log V) worst-case time. In spite Fibonacci heap is theoretically dominant in time efficiency, it might not work much better empirically with real-world data, because although large and intricate, the real-world road network graphs are sparse and the graph topology greatly affects the performance of the algorithms. SGO algorithm, a more efficient approch to address single source shortest path problem, is applied to solve optimal path planning problem in automotive navigation applications. Each road segment of the road graph leads to a particular location thus will hold unique distance values all the time. The SGO calculates the total distance from start location to the end of the each road segment and evaluates shortest path based on these calculations. The SGO algorithm and the Fibonacci and Binary heap variants of Dijkstra’s algorithm were tested on real-world intercity and intra-city road network graphs to evaluate running time efficiency of the algorithms in automotive navigation systems. The experimental results show that the SGO algorithm outperformed both the variants. The empirical superiority of the SGO algorithm suggests its usefulness in automotive navigation systems.

A Branch-Price-and-Cut algorithm for the Multi-Commodity two-echelon Distribution Problem

In the Multi-Commodity two-echelon Distribution Problem (MC2DP), multiple commodities are distributed in a two-echelon distribution system involving suppliers, distribution centres and customers. Each supplier may provide different commodities and each customer may request several commodities as well. In the first echelon, capacitated vehicles perform direct trips to transport the commodities from the suppliers to the distribution centres for consolidation purposes. In the second echelon, each distribution centre owns a fleet of capacitated vehicles to deliver the commodities to the customers through multi-stop routes. Commodities are compatible, i.e., they can be mixed in the vehicles. Finally, customer requests can be split by commodities, that is, a customer can be visited by several vehicles, but the total amount of each commodity has to be delivered by a single vehicle. The aim of the MC2DP is to minimize the total transportation cost to satisfy customer demands.We propose a set covering formulation for the MC2DP where the exponential number of variables relates to the routes in the delivery echelon. We develop a Branch-Price-and-Cut algorithm (BPC) to solve the problem. The pricing problem results in solving an Elementary Shortest Path Problem with Resource Constraints (ESPPRC) per distribution centre. We tackle the ESPPRC with a label setting dynamic programming algorithm which incorporates ng-path relaxation and a bidirectional labelling search. Pricing heuristics are invoked to speed up the procedure. In addition, the formulation is strengthened by integrating capacity cuts and two families of valid inequalities specific for the multiple commodities aspect of the problem.Our approach solves to optimality 439 over the 736 benchmark instances from the literature. The optimality gap of the unsolved instances is 2.1%, on average.

Dijkstra’s algorithm for shortest path problem under hesitant fuzzy environment using different operator

Dijkstra’s algorithm for shortest path problem under hesitant fuzzy environment using different operator

Note on a Vertex Stability Radius in the Shortest Path Problem

Note on a Vertex Stability Radius in the Shortest Path Problem

SimPLE, a visuotactile method learned in simulation to precisely pick, localize, regrasp, and place objects.

Existing robotic systems have a tension between generality and precision. Deployed solutions for robotic manipulation tend to fall into the paradigm of one robot solving a single task, lacking "precise generalization," or the ability to solve many tasks without compromising on precision. This paper explores solutions for precise and general pick and place. In precise pick and place, or kitting, the robot transforms an unstructured arrangement of objects into an organized arrangement, which can facilitate further manipulation. We propose SimPLE (Simulation to Pick Localize and placE) as a solution to precise pick and place. SimPLE learns to pick, regrasp, and place objects given the object's computer-aided design model and no prior experience. We developed three main components: task-aware grasping, visuotactile perception, and regrasp planning. Task-aware grasping computes affordances of grasps that are stable, observable, and favorable to placing. The visuotactile perception model relies on matching real observations against a set of simulated ones through supervised learning to estimate a distribution of likely object poses. Last, we computed a multistep pick-and-place plan by solving a shortest-path problem on a graph of hand-to-hand regrasps. On a dual-arm robot equipped with visuotactile sensing, SimPLE demonstrated pick and place of 15 diverse objects. The objects spanned a wide range of shapes, and SimPLE achieved successful placements into structured arrangements with 1-mm clearance more than 90% of the time for six objects and more than 80% of the time for 11 objects.

Partial dominance in branch-price-and-cut algorithms for vehicle routing and scheduling problems with a single-segment tradeoff

For many variants of vehicle routing and scheduling problems solved by a branch-price-and-cut (BPC) algorithm, the pricing subproblem is an elementary shortest-path problem with resource constraints (SPPRC) typically solved by a dynamic-programming labeling algorithm. Solving the SPPRC subproblems consumes most of the total BPC computation time. Critical to the performance of the labeling algorithms and thus the BPC algorithm as a whole is the use of effective dominance rules. Classical dominance rules rely on a pairwise comparison of labels and have been used in many labeling algorithms. In contrast, partial dominance describes situations where several labels together are needed to dominate another label, which can then be safely discarded. In this work, we consider SPPRCs, where a linear tradeoff describes the relationship between two resources. We derive a unified partial dominance rule to be used in ad hoc labeling algorithms for solving such SPPRCs as well as insights into its practical implementation. We introduce partial dominance for two important variants of the vehicle routing problem, namely the electric vehicle routing problem with time windows with a partial recharge policy and the split-delivery vehicle routing problem with time windows (SDVRPTW). Computational experiments show the effectiveness of the approach, in particular for the SDVRPTW, leading to an average reduction of 20% of the total BPC computation time, with savings of 30% for the more difficult instances requiring more than 600 s of computation time.

Reliable lifelong planning A*: Technique for re-optimizing reliable shortest paths when travel time distribution updating

Re-optimization technique is an efficient approach for solving the shortest path problem in dynamic deterministic networks, where link travel times are updated in real-time. However, existing re-optimization techniques, built on the assumption that link travel times are deterministic, cannot be used to solve reliable shortest path problems in real road networks with noticeable levels of travel time uncertainties. This study proposes a novel re-optimization technique, named reliable lifelong planning A* (RLPA*), for re-optimizing reliable shortest path finding results in dynamic stochastic networks, where link travel time distributions are updated in real-time. The proposed RLPA* technique can efficiently determine the optimal solution in dynamic stochastic networks by reusing path search results produced in the previous time instance. The proposed RLPA* technique is further utilized to solve the K reliable shortest paths problem, which is regarded as a series of reliable shortest path searches in a dynamic stochastic network. To validate the proposed algorithms, a comprehensive case study using real traffic data is conducted. The case study results demonstrated that the proposed algorithms significantly outperform the corresponding state-of-the-art algorithms on all testing networks.

Obtaining the optimal shortest path between two points on a quasi-developable Bézier-type surface using the Geodesic-based Q-learning algorithm

One of the most important problems is the computation of the shortest path between two points, because the search for the shortest path is very important in various fields, such as path planning of autonomous mobile robots. With this goal in mind, this study presents a novel method for determining the most efficient route connecting two locations on a surface. This approach relies on the geodesic properties of the surface points, the surface’s ability to be unfolded without distortion, and the utilization of the Q-learning algorithm.The first part of the proposed method constructs a quasi-developable Bézier-type surface from two boundary cubic trigonometric Bézier-type curves using the computed optimal four shape parameters. The subsequent stage of the introduced technique identifies the best and most direct route between two designated points on the derived quasi-developable Bézier-style surface. This is achieved through the utilization of the Geodesic-based Improved Epsilon GreedyQ-Learning (IEGQL) algorithm.Additionally, we have used the proposed algorithm in a mobile application developed for the elderly. This application involves offering a mobile game designed to ascertain the shortest path, with the goal of enhancing cognitive engagement among the elderly population as a preventive measure against Alzheimer’s disease. Moreover, the proposed method is compared with Dijkstra’s algorithm and A-Star algorithm, which are the best known algorithms for the shortest path problem.

Metarouting with automatic tunneling in multilayer networks

Metarouting allows for the modeling of routing protocols using an algebraic structure called routing algebra. Routing protocols requiring design or validation can easily be modeled using this approach. To date, however, existing research on routing algebras has mainly focused on applying this approach to routing protocols that are generally used in networks which have a single addressing and forwarding protocol. The basic algebraic structures used in such contexts are semirings, Sobrinho’s algebras and algebras of endomorphisms. In this paper, we propose the modification of these existing routing algebras to deal with networks that contain multiple forwarding protocols where tunnels are omnipresent. To achieve this, we define new algebraic structures derived from the three aforementioned ones, in order to model the generalized routing problem with automatic tunneling entitled valid paths algebra. All of our routing algebras are defined as semi-direct products of two structures, the well-known shortest paths algebra and the proposed valid paths algebra. These new algebras are isotonic and non-monotonic with a partial order. We propose a fixed point for those new algebras and we prove the iterative convergence to the optimal solution of the valid shortest paths problem.