Path Problem Research Articles

Node-to-Set Disjoint Paths Problem in Cross-Cubes

Node-to-Set Disjoint Paths Problem in Cross-Cubes

Solving fuzzy shortest path problem with vertex transfer penalties under type-2 fuzzy environment

Solving fuzzy shortest path problem with vertex transfer penalties under type-2 fuzzy environment

Enhanced bi-directional dynamic programming algorithm for the resource constrained shortest path problem

In this paper we propose an enhanced bi-directional dynamic programming algorithm for the Resource Constrained Shortest Path, Righini and Salani (2006). In particular we study the join procedure, one of the time consuming components of the algorithm, and propose a dominance based join that uses binary search and Pareto frontier exploration. The proposed method can be used in both node-join and arc-join procedures. The computational experiments are performed on two classes of problems both describing complete graphs with negative cost cycles. On the first set, with a single resource consumption, the proposed procedure slightly improves the overall computational time with savings up to 8.4% for the B class and just 0.4% for the most time consuming class M. On the second set, with multiple resources consumption, the proposed procedure exhibits a remarkable improvement with an average speedup of 39.2%.

Optimizing the integration of express bus services with micro-mobility: a case study

Considering a bus line, the problem of defining an express bus line consists in determining which stops of the original line will belong to the express line and the frequency of the new line. The objective of the optimization is a trade-off between the advantages of the passengers that benefit from the new service and the disadvantages of the passengers that cannot exploit the improved service. The advantages and disadvantages are measured in terms of total travel and waiting time.In this work we consider the introduction of micro-mobility, such as shared electric scooters or bikes, to mitigate the disadvantages and to increase the ridership of the express line. The contribution of the paper is twofold. On the one hand, we propose an improved mathematical model for defining an express bus line. The model is based on a longest path problem on an acyclic graph; thus, it is very efficient with respect to the other models in the literature. On the other hand, we consider the real case of two urban bus lines in the city of Milan and present the results computed on real passenger data. The results provide evidence that implementing an express line can improve total travel and waiting time. The time wasted by switching lines is in general higher than the time gained by the passengers. However, this can be compensated by the introduction of micro-mobility means of transportation.

Generated admissible orders for intervals by matrices and continuous functions

In this paper, we systematically study the algebraic structures of the spaces L([0,1]), encompassing all closed subintervals of [0,1], under the generated admissible orders. We first prove that the admissible order on L([0,1]) generated by a non-degenerate matrix must be the form of two weighted averaging operators. As a corollary, we deduce that each admissible order on L([0,1]) generated by a non-degenerate matrix and the standard order ≤ on [0,1] are not isomorphic. Furthermore, we show that each admissible order on L([0,1]) derived from two continuous mappings and the standard order ≤ on [0,1] are not isomorphic, partially answering a conjecture proposed by Santana et al. (2020) [38]. Besides, we prove that L([0,1]) is a complete lattice under the admissible order generated by two continuous mappings. This is the first result regarding the completeness of L([0,1]). Finally, we apply the admissible orders to solve a minimal path problem within the context of interval-valued fuzzy weighted graph. The above results theoretically refine the study of the classification, non-isomorphism, and completeness of admissible orders, while expanding the scope of interval-valued fuzzy sets in practical applications.

FBi-RRT: a path planning algorithm for manipulators with heuristic node expansion

AbstractThis paper is concerned with the problem of collision-free path planning for manipulators in multi-obstacle scenarios. Aiming at overcoming the deficiencies of existing algorithms in excessive time consumption and poor expansion quality, a path planning algorithm named Fast Bi-directional Rapidly-exploring Random Tree (FBi-RRT) with novel heuristic node expansion is proposed, which includes a selective-expansion strategy and a vertical-exploration strategy. The selective-expansion strategy is designed to guide the selection of the nearest-neighbor node to avoid the repeated expansion failure, thereby shortening the overall planning time. Also, the vertical-exploration strategy is developed to regulate the expansion direction of the collision nodes to escape from the obstacle space with less blindness, thus improving the expansion quality and further reducing time cost. Compared with previous planning algorithms, FBi-RRT can generate a feasible path for manipulators in a drastically shorter time. To validate the effectiveness of the proposed heuristic node expansion, FBi-RRT is conducted on a 6-DOF manipulator and tested in five scenarios. The experimental results demonstrate that FBi-RRT outperforms the existing methods in time consumption and expansion quality.

Spatial Mapping of Polling Units using Open-Source Software

The feasibility for adopting emerging Open-Source software and free mobile GIS data collection tools for Polling Unit mapping (PUs) and inventory was tested. The dynamism of KOBO COLLECT and QGIS Open-Source technologies were used in mapping and analyzing the spatial distribution of polling units in Eket Local Government Area. The study maximized the potentials of Open-Source technologies to provide a graphical inventory of polling units and area coverage for information management in the electioneering process. Spatial analysis tool of QGIS was used for solving shortest path problem centered around emergency response routing scenarios based on travel time and distance impedance from an origin to a destination. It also helped to portray the distribution within the study area in a bid to reveal pattern and accessibility to the populace. Results obtained will allow citizens and electorates access information on the locations of various polling units, which ultimately serve as a decision supporting system that would enhance their participation. The maps are also useful for future distribution of polling booths, security personnel, for solving transportation issues and in service delivery during election periods.

Finding the k Shortest Simple Paths: Time and Space Trade-offs

The k shortest simple path problem ( k SSP) asks to compute a set of top- k shortest simple paths from a source to a sink in a digraph. Yen (1971) proposed an algorithm with the best-known polynomial time complexity for this problem. Since then, the problem has been widely studied from an algorithm engineering perspective. The most noticeable proposals are the node-classification (NC) algorithm (Feng, 2014) and the sidetracks-based (SB) algorithm (Kurz, Mutzel, 2016). The latest offers the best running time at the price of a significant memory consumption. We first show how to speed up the SB algorithm using dynamic updates of shortest path trees resulting in a faster algorithm (SB*) with the same memory consumption. We then propose the parsimonious SB (PSB) algorithm that significantly reduces the memory consumption of SB at the cost of a small increase of the running time. Furthermore, we propose the postponed node-classification (PNC) algorithm that combines the best of the NC and the SB algorithms. It offers a significant speed up compared to the SB algorithm while using the same amount of memory as the NC algorithm. Our experimental results on complex networks show that all the considered algorithms have low memory consumption, and that the PSB algorithm is the fastest. On road networks, the relative performances of the algorithms depend on the number k of requested paths. Indeed, when the number k of requested paths is small (i.e., k ≤ 20 in our experiments), the SB* algorithm is the fastest among the considered algorithms, but it suffers from a large memory consumption and it offers very bad performances on some queries. When the number of requested paths is large (i.e., larger than 20 according to our experiments), the PNC algorithm is the fastest among the considered algorithms on road networks and it has a low memory footprint. The PNC algorithm is therefore a better choice on road networks.

Valid inequalities for the k-Color Shortest Path Problem

Given a digraph D=(V,A) where each arc (i,j)∈A has a cost dij∈R+ and a color c(i,j), a positive integer k, and vertices s,t∈V, the k-Color Shortest Path Problem consists in finding a path from s to t of minimum cost while using at most k distinct arc colors. We propose valid inequalities for the problem that proved to strengthen the linear relaxation of an existing Integer Linear Programming formulation for the problem. One exponential set of valid inequalities defines a new formulation for the problem that is solved by using a branch-and-cut algorithm. We introduce more challenging instances for the problem and present numerical experiments for both the benchmark and the new instances. Finally, we evaluate the individual and the collective use of the valid inequalities. Computational results for the proposed ideas and for existing solution approaches for the problem showed the effectiveness of the new inequalities in handling the new instances, both in terms of execution times and improvement of the linear relaxed solutions.

ERA*: Enhanced Relaxed A* algorithm for solving the shortest path problem in regular grid maps

This paper introduces a novel algorithm for solving the point-to-point shortest path problem in a static regular 8-neighbor connectivity (G8) grid. This algorithm can be seen as a generalization of Hadlock algorithm to G8 grids, and is shown to be theoretically equivalent to the relaxed A⁎ (RA⁎) algorithm in terms of the provided solution's path length, but with substantial time and memory savings, due to a completely different computation strategy, based on defining a set of lookup matrices. Through an experimental study on grid maps of various types and sizes (1290 runs on 43 maps), it is proven to be 2.25 times faster than RA⁎ and 17 times faster than the original A⁎, in average. Moreover, it is more memory-efficient, since it does not need to store a G score matrix.

Network Routing Optimization Using Tabu Search Algorithm in Dynamic Routing

Internet penetration is increasing along with the need for data packages for communication such as social media, chatting, video conferencing and others. On large-scale networks such as the Internet, dynamic routing is used to build routing protocol information in the routing table automatically. Currently, Djikstra's algorithm is used to solve the shortest path problem in dynamic routing. In this research, the optimization of the algorithm is carried out in determining the best path or trajectory. One of the optimization algorithms is the Tabu Search Algorithm which can guide heuristic local search procedures to explore the solution area outside the local optimum point. This optimization is assessed from the test parameters measured from the smallest cost. The data analyzed is in the form of bandwidth and topological flow. From the results of tracing the path of data packets sent through 9 routers using the Tabu Search algorithm with the parameters namely number of Neighbor Solutions = 50, Length of tabu list = 10, Maximum Number of Iterations = 100, the result of the path matrix value is 180.9676. The path taken is router 0-2-4-8-9

Path Planning for Unified Scheduling of Multi-Robot Based on BSO Algorithm

The technology for path planning of independent mobile robots is mature, but multi-robot path planning for unified scheduling and allocation is much more complex than single-robot path planning. This requires consideration of collision problems between robots, general optimal path problems, etc. This paper proposes the use of the BSO algorithm for unified scheduling and allocation of multiple robots to improve the efficiency of task execution. The BSO algorithm is a new type of intelligent optimization algorithm that uses clustering ideas to search for local optimal solutions and obtains global optimal solutions by comparing local optimal solutions. It also uses mutation ideas to increase the diversity of the algorithm and avoid becoming trapped in local optimal solutions. Using the GA/SA algorithm and the proposed BSO algorithm for computer simulation comparison, we obtained the optimal path planning for the three robots under unified scheduling. The total distance of the optimal path obtained by the BSO algorithm was 27.36% and 25.31% shorter than those of the GA and SA algorithms, respectively. To further test the performance of the BSO algorithm, we conducted additional experiments on the unified scheduling of multiple robots. The experimental results show that the proposed BSO algorithm can significantly improve the efficiency. The multi-robot under unified scheduling performs point-to-point path planning without collisions, and they can traverse all task target points in the shortest path without repetition. This algorithm is suitable for multi-robot tasks in large-scale environments.

Using a Node–Child Matrix to Address the Quickest Path Problem in Multistate Flow Networks under Transmission Cost Constraints

The quickest path problem in multistate flow networks, which is also known as the quickest path reliability problem (QPRP), aims at calculating the probability of successfully sending a minimum of d flow units/data/commodity from a source node to a destination node via one minimal path (MP) within a specified time frame of T units. Several exact and approximative algorithms have been proposed in the literature to address this problem. Most of the exact algorithms in the literature need prior knowledge of all of the network’s minimal paths (MPs), which is considered a weak point. In addition to the time, the budget is always limited in real-world systems, making it an essential consideration in the analysis of systems’ performance. Hence, this study considers the QPRP under cost constraints and provides an efficient approach based on a node–child matrix to address the problem without knowing the MPs. We show the correctness of the algorithm, compute the complexity results, illustrate it through a benchmark example, and describe our extensive experimental results on one thousand randomly generated test problems and well-established benchmarks to showcase its practical superiority over the available algorithms in the literature.

Research on global path planning of intelligent vehicles based on improved ant colony algorithm

According to the needs of intelligent vehicles, this paper proposes a global path planning algorithm based on improved ant colony algorithm to address the problems of slow convergence speed and multiple path bends in ant colony algorithm. By redefining the heuristic function of ant colony algorithm, adopting fallback and fill mechanisms, setting safe distances, modifying pheromone concentration update rules, and limiting the range of pheromone concentration, the ant colony algorithm is improved. And by deleting redundant points on the path, selecting key turning points and cubic B-spline curves for path smoothing, the planned path is shortest and smooth, which can meet the needs of path following. Through simulation, the improved ant colony algorithm has the advantages of fast convergence and smooth path.

SOZIL: Self-Optimal Zero-Shot Imitation Learning

Zero-shot imitation learning has demonstrated its superiority to learn complex robotic tasks with less human participation. Recent studies show convincing performance under the condition that the robot follows the demonstration strictly by the learned inverse model. However, these methods are difficult to achieve satisfactory performance in imitation when the demonstration is suboptimal, and the learning of the learned inverse models is vulnerable to label ambiguity issues. In this paper, we propose Self-Optimal Zero-shot Imitation Learning (SOZIL) to tackle these problems. The contribution of SOZIL is twofold. First, Goal Consistency Loss (GCL) is designed to learn the multi-step goal-conditioned policy from exploration data. By directly using the goal state as supervision, GCL solves the label ambiguity problem caused by trajectory and action diversity. Second, Estimation-based Keyframe Extraction (EKE) is developed to optimize demonstrations. We formulate the keyframe extraction process as a path optimization problem under suboptimal control. By predicting the performance of the learned policy in executing transitions of any two states, EKE creates a directed graph containing all candidate paths and extracts keyframes by solving the graph’s shortest path problem. Furthermore, the proposed method is evaluated with various simulated and real-world robotic manipulating experiments such as cable harness assembly, rope manipulation, and block moving. Experimental results show that SOZIL achieves a superior success rate and manipulation efficiency than baselines.

Dynamic User-Scheduling and Power Allocation for SWIPT Aided Federated Learning: A Deep Learning Approach

Federated learning (FL) has been considered as a promising paradigm for enabling distributed machine learning (ML) in wireless networks. To address the limited energy capacity of wireless devices, we propose a simultaneous wireless information and power transfer (SWIPT) aided FL, in which one FL server (FLS) co-located at a cellular base station (BS) uses SWIPT to simultaneously broadcast the global model to wireless user-devices (UDs) and provide wireless power transfer to them. The UDs then use the harvested energy to train their local models and further transmit the local models to the FLS for aggregation. To improve the spectrum efficiency, we consider that the UDs form a non-orthogonal multiple access (NOMA) group for simultaneously sending their local models over the same spectrum channel. Taking the UDs' time-varying available energy and channel conditions into account, we propose a dynamic optimization of the UDs-scheduling, the BS's transmit-power allocation, and the UDs' power-splitting factors for SWIPT, with the objective of minimizing the long-term energy consumption while ensuring the FL convergence. The optimization problem, however, is challenging to solve since it is a finite-horizon dynamic programming problem but with an unknown stopping time, and moreover, the action space covers both discrete and continuous variables. To address these difficulties, we first execute a series of equivalent transformations to reduce the number of decision variables and then formulate the problem as a stochastic shortest path problem, based on which we propose an actor-critic deep reinforcement learning algorithm with the proximal policy optimization to efficiently learn the policy that dynamically adjusts the UDs-scheduling for FL as well as the BS's transmit-power for SWIPT. Numerical results validate the effectiveness and performance of our proposed algorithm. The results demonstrate that our proposed algorithm can effectively reduce the long-term energy consumption in comparison with two baseline algorithms.

A Whirling Dervish: Polynomial-Time Algorithm for the Regional SRLG-Disjoint Paths Problem

The current best practice in survivable routing is to compute link or node disjoint paths in the network topology graph. It can protect single-point failures; however, several failure events may cause the interruption of multiple network elements. The set of network elements subject to potential failure events is called Shared Risk Link Group (SRLG), identified during network planning. Unfortunately, for any given list of SRLGs, finding two paths that can survive a single SRLG failure is NP-Complete. In this paper, we provide a polynomial-time SRLG-disjoint routing algorithm for planar network topologies and a large set of SRLGs. Namely, we focus on regional failures, where the failed network elements must not be far from each other. We use a flexible definition of regional failure, where the only restrictions are that i) the topology is a planar graph, ii) each SRLG forms a set of connected edges in the dual of the planar graph, and iii) for each node <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v$</tex-math> </inline-formula> , the links incident to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v$</tex-math> </inline-formula> are part of an SRLG. The proposed algorithm is based on a max-min theorem. Through extensive simulations, we show that the algorithm scales well with the network size, and one of the paths returned by the algorithm is only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4$</tex-math> </inline-formula> % longer than the shortest path on average.

A multiphase dynamic programming algorithm for the shortest path problem with resource constraints

The shortest path problem with resource constraints (SPPRC) finds a least cost path between two nodes in a network while respecting constraints on resource consumption. The problem is mainly used as a subproblem inside column generation for crew scheduling and vehicle routing problems. The standard approach for the subproblems is based on dynamic programming (DP). This class of methods is generally effective in practice when there are only a few resources, but it seems to be time-consuming for huge instances with many resources. To handle this issue, we propose a new exact primal algorithm called the multiphase dynamic programming algorithm (MPDPA) to solve the SPPRC in acyclic networks. The proposed approach splits the state-space into small disjoint subspaces. These subspaces are sequentially explored in several iterations, where each iteration builds on the previous ones, to reduce the dimension of the subspaces to explore and to quickly generate better paths. Computational experiments on vehicle and crew scheduling instances show the excellent performance of our approach compared to the standard DP method. On the one hand, MPDPA returns optimal solutions while achieving time reduction factors between 1.44 and 3.59 on average compared to DP. On the other hand, MPDPA is able to generate feasible paths with up to 90% of the optimal cost in less than 10% of the time required by standard DP. This feature is useful in column generation and may greatly reduce the computational effort, because we can stop the MPDPA solution process once columns with sufficiently negative reduced costs are obtained.

ERCA*: A New Approach for the Resource Constrained Shortest Path Problem

ERCA*: A New Approach for the Resource Constrained Shortest Path Problem

Multiagent, Multitarget Path Planning in Markov Decision Processes

Missions for autonomous systems often require agents to visit multiple targets in complex operating conditions. This work considers the problem of visiting a set of targets in minimum time by a team of non-communicating agents in a Markov decision process (MDP). The single-agent problem is at least NP-complete by reducing it to a Hamiltonian path problem. We first discuss an optimal algorithm based on Bellman's optimality equation that is exponential in the number of target states. Then, we trade-off optimality for time complexity by presenting a suboptimal algorithm that is polynomial at each time step. We prove that the proposed algorithm generates optimal policies for certain classes of MDPs. Extending our procedure to the multi-agent case, we propose a target partitioning algorithm that approximately minimizes the expected time to visit the targets. We prove that our algorithm generates optimal partitions for clustered target scenarios. We present the performance of our algorithms on random MDPs and gridworld environments inspired by ocean dynamics. We show that our algorithms are much faster than the optimal procedure and more optimal than the currently available heuristic.