As manufacturing progresses, the demand for precision inspection of complex parts has intensified. To guarantee functionality and sensory performance, high-efficiency 3D shape measurement is required. In this paper, a collaborative robot-based approach for efficient and high-precision 3D shape inspection of complex parts is proposed. The system employs a collaborative robot to drive the scanner along optimized trajectories. First, the configuration of the inspection system is presented, and the ideal measurement mode for the sensor is analyzed. Subsequently, adaptive viewpoints are generated through parametric discretization based on surface geometric features. For inter-region scanning path planning, the problem is modeled as the Shortest Path Problem (SPP) within the framework of the Traveling Salesman Problem (TSP) and solved by constructing a Successive Approximation Algorithm (SAA). Furthermore, a Modified Denavit-Hartenberg (MDH) method is applied to establish the precise kinematic model of the collaborative robot. Inverse kinematics solutions are derived to convert planned viewpoints into target joint configurations, thereby achieving precise end-effector pose control. Simulation and experimental results on an engine cover and a cylinder head demonstrate that the proposed approach enables comprehensive 3D shape inspection of complex parts in a single setup and achieves higher efficiency and accuracy compared to existing methods. This work offers a viable solution for integrating robotic actuation and active sensing in the automated inspection of complex geometries.

The shortest path problem which can not be solved by classical Dijkstra algorithm and Moore-Bellman-Ford algorithm appears frequently, for example, the Anti-risk Path Problem proposed by Xiao et al. To address this kind of shortest path problem, the present work proposes and studies a general single-source shortest path problem, which is motivated by current interest in needing to extend the total weight function of paths on a network and the classical shortest path problem. Firstly, define the path functional on a set of certain paths with same source on a graph; introduce a few concepts of the defined path functional; and make some discussions on the properties of the path functional. Secondly, develop a kind of general single-source shortest path problem (GSSSP). Thirdly, following respectively the approaches of the well known Dijkstra’s algorithm and Moore-Bellman-Ford algorithm, design an extended Dijkstra’s algorithm (EDA) and an extended Moore-Bellman-Ford algorithm (EMBFA) to solve the problem GSSSP under certain given conditions. Fourthly, under the assumption that the value of related path functional for any path can be obtained in M(n) time, prove respectively the algorithm EDA solving the problem GSSSP in O(n2)M(n) time and the algorithm EMBFA solving the problem GSSSP in O(mn)M(n) time. Finally, some applications of the designed algorithms are shown with a few examples. What we done can not only improve both the researches and the applications of the shortest path theory, but also promote the development of the researches and the applications of other combinatorial optimization problems, promote the development of the algorithm theory and promote the development of the artificial intelligence.

The air-rail inter-modal transport is a feasible choice to enlarge the freight service scope of high-speed railway. Essentially, optimising the service plan for high-speed rail express under inter-modal mainly involves determining the train and flight trips, and space-time route selections for each batch of express shipment from the origin to the destination. We construct an extended space-time network to capture the transport and transfer space-time attributes of the serviced express shipments. A multi-commodity flow model is then established with a series of practical constraints. The Lagrangian relaxation algorithm is designed to decompose the original problem into a shortest path problem of a single-batch express shipment in a multi-dimensional network. The sub-problem is solved by the dynamic programming method, and a heuristic algorithm based on sub-gradient sorting is designed to ensure the feasibility of the dual solutions. In order to compare the performance of the traditional solver method with that of the LR algorithm, a nonlinear mixed-integer programming model was constructed in the appendix and solved by using the DICOPT solver. Taking the Shanghai-Kunming corridor as an example, the experimental results demonstrate that the LR algorithm can obtain high-quality solutions within a relatively short time, while the traditional solver method has certain limitations. Furthermore, the inter-modal transport is validated with a significant advantage in expanding the service scope of express demand. The research results are of great theoretical significance for the rational allocation of transportation resources and enhancement of the quality and efficiency of high-speed rail express services.

ABSTRACT Shortest path problems often arise in contexts where travel times are uncertain. In these settings, reliable paths are often valued more than paths with lower expected travel times. This has led to several variants of reliable shortest path problems (RSPP) that handle travel time reliability differently. We propose an algorithmic framework for solving RSPPs with non‐negatively correlated travel times and resource constraints. By building upon the flexibility of the pulse algorithm, our unified and exact algorithmic framework solves multiple variants of the RSPP: the ‐reliable shortest path (‐RSP), the maximum probability of on‐time arrival (MPOAP) problem, and the shortest ‐reliable path (S‐). We derive a bound on the reliability of path travel times and incorporate three pruning strategies: bounds, infeasibility, and dominance, leveraging properties of the normal distribution and non‐negative correlation structures. Computational experiments on large‐scale transportation networks (with up to 33 113 nodes and 75 379 arcs) demonstrate that the framework achieves a ten‐fold speed improvement over state‐of‐the‐art methods, highlighting its potential real‐world applications and extensions to related problems.

Previous studies, constrained by the overly rigid stability requirements, often fail to adapt to complex systems and struggle to identify stable outcomes that align with the practical context of multi-agent resource allocation. To address the three-sided matching problem in complex socio-technical and business management systems, this paper proposes a fuzzy stable matching method for three-sided agents under a framework of combinatorial preference relations, integrating network and decision theory. First, we construct a membership function to measure the degree of preference satisfaction between elements of different agents, and then define the concept of fuzzy stability. By incorporating preference satisfaction, we introduce the notion of fuzzy blocking strength and derive the generation conditions for blocking triples and fuzzy stability under the fuzzy stable criterion. Furthermore, we abstract the three-sided matching problem with combined preference relations into a shortest path problem. Second, we prove the equivalence between the shortest path solution and the stable matching outcome. We adopt Dijkstra’s algorithm for problem-solving and derive the time complexity of the algorithm under the pruning strategy. Finally, we apply the proposed model and algorithm to a case study of project assignment in software companies, thereby verifying the feasibility and effectiveness of this three-sided matching method. Compared with existing approaches, the fuzzy stable matching method developed in this study demonstrates distinct advantages in handling preference uncertainty and system complexity. It provides a more universal theoretical tool and computational approach for solving flexible resource allocation problems prevalent in real-world scenarios.

In this paper, the shortest path problem is considered with emphasis on reducing the COVID‐19 contamination risk. Generally, the shortest path problem focuses on planning the minimal path considering the path distance criterion, and here the aim is to generate the shortest safe path with obstacles free collision and virus infection saving. For that, the novel Dhouib‐Matrix Shortest Path Problem (DM‐SPP) method is enhanced to reduce the probability of catching COVID‐19 by keeping people away from crowded and risked space. DM‐SPP is enriched with a grid map, namely, the risk pandemic grid map, gathering the human flow density of each area in order to mark the risk epidemic zone. To prove the performance of DM‐SPP to optimize the trajectory in COVID‐19 virus infection, two case sites are used (a campus case study represented as 40 × 40 grid map and an experimental platform of 50 × 50 grid map). The solutions generated by DM‐SPP are graphically represented using the Python programing language, and its results are compared to the results of recently developed metaheuristics in the literature, namely, the classical ant colony optimization metaheuristic, the improved ant colony optimization metaheuristic, the classical A ∗ algorithm, and the improved A ∗ algorithm.

Arctic route planning under ice uncertainty: A risk-averse stochastic shortest path problem

The centrality of a node is often used to measure its importance to the structure of a network. Some centrality measures can be extended to measure the importance of a path. In this paper, we consider the problem of finding the most central shortest path. We show that the computational complexity of this problem depends on the measure of centrality used and in the case of degree centrality, whether the network is weighted or not. We develop a polynomial algorithm for the most degree-central shortest path problem with the worst-case running time of [Formula: see text], where | V | is the number of vertices, | E | is the number of edges, and [Formula: see text] is the maximum degree of the graph. In addition, we show that the same problem is NP-hard on a weighted graph. Furthermore, we show that the problem of finding the most betweenness-central shortest path is solvable in polynomial time, whereas finding the most closeness-central shortest path is NP-hard, regardless of whether the graph is weighted or not. We also develop an algorithm for finding the most betweenness-central shortest path with a running time of [Formula: see text] on unweighted graphs and [Formula: see text] on graphs with positively weighted edges. To conclude our paper, we conduct a numerical study of our algorithms on synthetic and real-world networks and compare our results with the existing literature. History: Accepted by Russell Bent, Area Editor for Network Optimization: Algorithms and Applications. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0945 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0945 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

The Floyd–Warshall algorithm, which uses a classic dynamic programming approach, provides a solution to the all-pairs shortest paths problem. However, for sparse graphs, iteratively applying Dijkstra’s, or some other similar algorithm from each node, often proves to be more efficient. We introduce a novel technique based on a structural decomposition of the input graph into strongly connected components, allowing us to exploit the disconnectedness of the graph by avoiding redundant relaxation attempts on nodes that are not reachable from the source component. Using an empirical evaluation, where execution time is measured, we demonstrate that our approach outperforms existing alternatives on disconnected graphs.

In this paper, we address a combinatorial optimization problem arising when deciding the routes to be followed by several autonomous vehicles. More particularly, each available autonomous vehicle needs to travel from its origin to its destination (e.g. to deliver or collect material). Considering all the constraints involved, the objective consists of minimizing the time at which the last vehicle reaches its destination. This problem is denoted as the Concurrent Shortest Path Problem (CSPP), which is a generalization of the Disjoint Shortest Path Problem existing in the literature. We formulate the CSPP using two Mixed Integer Linear Programming models: one based on flow formulations and another based on bin-packing formulations. Due to the high complexity of the problem, neither of these models can solve large instances of the CSPP. Therefore, two heuristic approaches are also proposed. Computational experiments over randomly generated instances empirically prove that the heuristic approaches yield good-quality solutions in short CPU times, enabling real-time management of autonomous fleets in smart-city and industrial environments.

There are real-world complex networks formed by people, roads, communication network nodes, genes, file Servers and financial transactions etc based on their interdependent associations. Often, there exist multiple paths to connect one of the network nodes to another in the networks. Various situations demand shortest path to reach from one node to another. To find shortest path between a pair of the nodes is challenging in terms of computational cost, therefore, competing algorithms for Shortest Path problem are analyzed to predict their greediness for the computing resources such as memory, bandwidth, or hardware but most often its computational time. The scope of this work is to provide comprehensive knowledge on single-source shortest path problem and analyze time complexity of the prominent single-source shortest path algorithms.

In the field of research on Deoxyribonucleic acid (DNA), graph theory can be applied to model the structure of a DNA molecule. In particular, the shortest path problem in graph theory can be used to identify the shortest path between vertices of a graph. Hence, it is possible to apply the shortest path problem for minimization of a DNA string so that the time taken for computation of genome assembly can be reduced. This paper presents a method to represent a DNA string graphically where the shortest path is calculated for the graph generated from the DNA string, and the shortest path is then used to minimize the DNA string. In this research, a DNA string is presented in graphical form by using base pairs of length two as the vertices where the initial bases used are Adenine (A) and Guanine (G) which are the main bases in purine. The number of base pairs between adjacent vertices in the DNA string is represented by the edges. The graph is then reduced by following a given set of rules where the shortest path is calculated for all start and end vertices of the reduced graph. Next, the simplification of the graph is done based on the shortest paths obtained by removing all the untraversed paths where the Euler path for the simplified graph is used to form a minimized DNA string. The result shows that a minimized DNA string can be obtained by simplifying the graph of the DNA string using the shortest path problem

Since its inception in 1959, Dijkstra's algorithm has long been regarded as the optimal solution for graph theory's single-source shortest path problem. Over the past seven decades, it has permeated daily life, underpinning applications in traffic navigation, geographic information systems (GIS), robotics, and communication networks. However, amid rapid technological advancement, the rise of smart cities and real-time systems has exposed critical limitations of traditional approachesnamely, inadequate scalability, poor dynamic adaptability, and exponential degradation in computational efficiency with increasing scale. Leveraging extensive literature, this paper systematically analyzes the algorithm's evolution across five dimensions: static optimizationheuristic extensiondynamic update3D integrationengineering deployment. The salient results encompass: refined data structures, such as Net-Arc dual-table compression, facilitating O(1) adjacency-edge access; augmented heuristic algorithms that markedly improve search efficacy while preserving solution optimality (or with negligible concession); dynamic weight update protocols diminishing computational complexity to O(|Q| log|Q|); the incorporation of three-dimensional modeling, enhancing Unmanned Aerial Vehicle (UAV) ranging precision by 8.72 meters; and pragmatic implementations, exemplified by optimal railway route determination. Future research must address negative-weight edges, distributed computing for ultra-large graphs (>10 nodes), and real-time path planning integrated with digital twins.

Abstract: Early Indian trade routes It was critical to the economic and cultural development of the subcontinent. These intricate networks unified the cities, ports, and trade centers of the Indus valley to the Southeast Asia and have enabled the exchange of goods including spices, textile, metals, and medicinal plants. Modern Operations Research (OR) tools have been also employed in this study: The graph theory and shortest path problem will be used in order to analyze and model those historical trade routes. Historical sources, archaeological data, geographical reconstructions have been used to model ancient trade networks as weighted graphs (the nodes are trade centers, and the edges are routes with related distance, and risks e.g., terrain difficulty, political instability, banditry, and weather hazards). Applying such algorithms as Dijkstra’s, Bellman-Ford, and Floyd-Warshall the study is able to find the optimal paths, which might have been favoured by ancient traders, on various constraints, namely, travel time, cost, and risk. The multi-objective optimization model is also presented in the paper to consider efficiency and safety in order to capture the real-life decisions taken by traders in dynamic historical situations. The results point to the existence of proto-optimization behavior in ancient Indian trade and they offer a new interdisciplinary solution on how to relate historical geography and mathematical modeling. This study does not only unearth the strategic genius of the ancient Indian traders but also proves the evergreen applicability of OR in resolving practical issues.

The development of the computing power network has brought about a revolutionary effect on network routing architecture. As a result, the computing-aware network routing problem has been raised to explore routing various computational tasks to appropriate computing resources in the dynamic network. In this study, we propose a heuristic-based computing-aware routing algorithm to achieve the optimal routing path by considering the dynamic network performance and computing resource status simultaneously. Our proposed approach models the dynamic network using time-varying node and edge weights, which are obtained by mapping basic performance indicators to weights according to quality-of-service requirements. This allows us to improve the user’s experience more effectively during the routing process. Moreover, a novel heuristic-based algorithm, which creatively transforms the computing-aware routing problem into a single-source shortest path problem, has been designed to achieve the comprehensive optimal routing path. The experimental results, based on both simulated networks and a real dedicated network in Zhejiang, demonstrate that our proposed method can obtain the comprehensive optimal routing path with a lower computing time cost than enumerating search. Furthermore, our proposed computing-aware routing method has been proven to be robust to the dynamics of the network, computing resources, and service load changes.

We present a novel algorithm that fuses the existing convex-programming based approach with heuristic information to find optimality guarantees and near-optimal paths for the Shortest Path Problem in the Graph of Convex Sets (SPP-GCS). Our method, inspired by A* initiates a best-first-like procedure from a designated subset of vertices and iteratively expands it until further growth is neither possible nor beneficial. Traditionally, obtaining solutions with bounds for an optimization problem involves solving a relaxation, modifying the relaxed solution to a feasible one, and then comparing the two solutions to establish bounds. However, for SPP-GCS, we demonstrate that reversing this process can be more advantageous, especially with Euclidean travel costs. In other words, we initially employ A* to find a feasible solution for SPP-GCS, then solve a convex relaxation restricted to the vertices explored by A* to obtain a relaxed solution, and finally, compare the solutions to derive bounds. We present numerical results to highlight the advantages of our algorithm over the existing approach in terms of the sizes of the convex programs solved and computation time.

Abstract Partial inverse combinatorial optimization problems are bilevel optimization problems in which the leader aims to incentivize the follower to include respectively not include given sets of elements in the solution of their combinatorial problem. If the sets of required and forbidden elements define a complete follower solution and the follower problem is solvable in polynomial time, then the inverse combinatorial problem is also solvable in polynomial time. In contrast, partial inverse problems can be NP-complete when the follower problem is solvable in polynomial time. This applies e.g. to the partial inverse min cut problem. In this paper, we consider partial inverse combinatorial optimization problems in which weights can only be increased. Furthermore, we assume that the lower-level combinatorial problem can be solved as a linear program. In this setting, we show that the partial inverse shortest path problem on a directed acyclic graph is NP-complete. Moreover, the partial inverse assignment problem is NP-complete. Both results even hold if there is only one required arc or edge, respectively. For solving partial inverse combinatorial optimization problems with only weight increases, we present a novel branch-and-bound scheme that exploits the difference in complexity between complete inverse and partial inverse versions of a problem. For both primal heuristics and node relaxations, we use auxiliary problems that are basically complete inverse problems on similar instances. Branching is done on follower variables. We test our approach on partial inverse shortest path, assignment and min cut problems, and computationally compare it to an MPCC reformulation as well as a decomposition scheme.

ABSTRACTWe introduce a new bilevel version of the classic shortest path problem and completely characterize its computational complexity with respect to several problem variants. In our problem, the leader and the follower each control a subset of the edges of a graph and together aim at building a path between two given vertices, while each of the two players minimizes the cost of the resulting path according to their own cost function. We investigate both directed and undirected graphs, as well as the special case of directed acyclic graphs. Moreover, we distinguish two versions of the follower's problem: Either they have to complete the edge set selected by the leader such that the joint solution is exactly a path or they have to complete the edge set selected by the leader such that the joint solution is a superset of a path. In general, the bilevel problem turns out to be much harder in the former case: We show that the follower's problem is already NP‐hard here and that the leader's problem is even hard for the second level of the polynomial hierarchy, while both problems are one level easier in the latter case. Interestingly, for directed acyclic graphs, this difference turns around, as we give a polynomial‐time algorithm for the first version of the bilevel problem, but it stays NP‐hard in the second case. Finally, we consider restrictions that render the problem tractable. We prove that, for a constant number of leader's edges, one of our problem variants is actually equivalent to the shortest‐‐cycle problem, which is a known combinatorial problem with partially unresolved complexity status. In particular, our problem admits a polynomial‐time randomized algorithm that can be derandomized if and only if the shortest‐‐cycle problem admits a deterministic polynomial‐time algorithm.

Given a set of items, each requiring a set of elements, the set-union bin packing problem (SUBP) consists of grouping all items into a minimum number of bins such that each item is assigned to exactly one bin and the total weight of all distinct elements required in a bin does not exceed its capacity. The SUBP is a generalization of the well-known bin-packing problem, where items can share one or more elements in a nonadditive fashion. In the literature, it has been addressed by various names such as pagination problem, job grouping problem, tool switching problem, or bin packing problem with overlapping items. We propose a branch-and-price (B&P) algorithm for solving the SUBP. For the column-generation pricing problem, which is a set-union knapsack problem (SUKP), we present and explore alternative solution methods, namely the direct solution of an integer program with a general-purpose MIP solver and two labeling algorithms on ad hoc defined graphs. The overall best B&P variant combines an upfront greedy pricing heuristic and an item-based labeling approach without the application of any dominance. The latter is based on the representation of the pricing problem as a shortest path problem with resource constraints and relies on strong completion bounds as acceleration technique. Ryan-and-Foster branching is applied to ensure integer solutions. Extensive computational results demonstrate the effectiveness of the proposed method. Our B&P significantly outperforms the state-of-the-art IP formulations. It solves to optimality more than 10,000 instances from the literature that have only been solved heuristically before, improving the best known solutions for more than half of the benchmark. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by Deutsche Forschungsgemeinschaft [Grant GS 83/1-1 Project No. 418727865]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0791 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0791 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Bidirectional heuristic search has the potential to decrease search time in combinatorial search problems amenable to backward search. To date, bidirectional search has been limited to minimization or shortest path problems. This paper extends the notion of bidirectional heuristic search to (constrained) longest path problems, which turns out to be non-trivial due to the path necessarily being part of the state and the inapplicability of standard bidirectional heuristic search techniques such as meet-in-the-middle (MM) and BAE*. We present a basic bidirectional heuristic search for longest simple path (LSP) in undirected graphs, and prove its correctness. We then suggest several refinements and optimizations, as well as a generalization to other types of longest path problems Coil-in-a-box (CIB). Empirical evaluation shows that, as with many forms of bidirectional search, sometimes unidirectional search wins, but for a sizable chunk of problem instance types, bidirectional search performs better by expanding fewer nodes and achieves a shorter runtime despite the increased overhead per expansion.