Shortest Path Problem Research Articles

Revolutionizing energy infrastructure: Automated route planning for underground transmission lines in Phnom Penh

The capacity of conventional overhead transmission lines (OTL) has been exceeded by the energy consumption of Cambodia's flourishing cities. Underground transmission lines (UTL) provide a unique solution due to their technical and environmental advantages. However, engineers must contend with high UTL costs and the need for effective conduit cable installation. This study presents a solution that automates the shortest route method for determining the optimal path for routing UTL power supply cables between any two substations. Multiple algorithms are evaluated to identify the most effective solution to this design's inherent shortest-path problem. Using Geographic Information System (GIS) data, potential UTL network routes in Phnom Penh, Cambodia can be investigated. The ultimate product is an automated route planning support tool that has the potential to revolutionize energy project planning by generating optimal routes for UTL. This will increase efficiency and lower costs, making UTL a more viable option for meeting the energy demands of modern infrastructure. This study contributes significantly to the field of energy infrastructure planning and has the potential to be replicated in cities around the world.

Curvature-constrained Steiner networks with three terminals

A procedure is presented for finding the shortest network connecting three given undirected points, subject to a curvature constraint on both the path joining two of the points and the path that connects to the third point. The problem is a generalisation of the Fermat–Torricelli problem and is related to a shortest curvature-constrained path problem that was solved by Dubins. The procedure has the potential to be applied to the optimal design of decline networks in underground mines.

Sparse randomized policies for Markov decision processes based on Tsallis divergence regularization

This work investigates a somewhat different point of view on Markov decision processes by reinterpreting them as a randomized shortest paths problem on a bipartite graph, therefore establishing bridges with entropy-regularized reinforcement learning. The graph structure contains the set of states as “left” nodes and the set of actions as “right” nodes. In that context, the action-to-state transition probabilities are provided by the environment whereas the state-to-action probabilities correspond to the (stochastic) policy to be found. The randomized shortest paths formalism (minimizing expected cost to the goal state subject to (Shannon or Tsallis) relative entropy regularization) is then readily applied to this bipartite structure, providing a possibly sparse stochastic policy interpolating between a least-cost and a purely random policy. The algorithm computing the policy is closely related to the dual linear programming formulation of the Markov decision processes to which the relative entropy regularization term, multiplied by a scaling factor balancing exploitation and exploration (the temperature), is added. It is derived from well-known techniques of discrete optimal control, relying on costates (Lagrange parameters) backward computation. In summary, the proposed algorithm allows the design of optimal stochastic – but still sparse – policies, ranging from a purely rational to a random behavior, depending on the temperature parameter.

A linear time algorithm for linearizing quadratic and higher-order shortest path problems

AbstractAn instance of the NP-hard Quadratic Shortest Path Problem (QSPP) is called linearizable iff it is equivalent to an instance of the classic Shortest Path Problem (SPP) on the same input digraph. The linearization problem for the QSPP (LinQSPP) decides whether a given QSPP instance is linearizable and determines the corresponding SPP instance in the positive case. We provide a novel linear time algorithm for the LinQSPP on acyclic digraphs which runs considerably faster than the previously best algorithm. The algorithm is based on a new insight revealing that the linearizability of the QSPP for acyclic digraphs can be seen as a local property. Our approach extends to the more general higher-order shortest path problem.

Stable source connection and assignment problems as multi-period shortest path problems

Stable source connection and assignment problems as multi-period shortest path problems

Labeling methods for partially ordered paths

The landscape of applications and subroutines relying on shortest path computations continues to grow steadily. This growth is driven by the undeniable success of shortest path algorithms in theory and practice. It also introduces new challenges as the models and assessing the optimality of paths become more complicated. Hence, multiple recent publications in the field adapt existing labeling methods in an ad hoc fashion to their specific problem variant without considering the underlying general structure: they always deal with multi-criteria scenarios, and those criteria define different partial orders on the paths. In this paper, we introduce the partial order shortest path problem (POSP), a generalization of the multi-objective shortest path problem (MOSP) and in turn also of the classical shortest path problem. POSP captures the particular structure of many shortest path applications as special cases. In this generality, we study optimality conditions or the lack of them, depending on the objective functions’ properties. Our final contribution is a big lookup table summarizing our findings and providing the reader with an easy way to choose among the most recent multi-criteria shortest path algorithms depending on their problems’ weight structure. Examples range from time-dependent shortest path bottleneck path problems to the electric vehicle shortest path problem with recharging and complex financial weight functions studied in the public transportation community. Our results hold for general digraphs and, therefore, surpass previous generalizations that were limited to acyclic graphs.

PSO Based Constraint Optimization of Intuitionistic Fuzzy Shortest Path Problem in an Undirected Network

PSO Based Constraint Optimization of Intuitionistic Fuzzy Shortest Path Problem in an Undirected Network

An application of the graph approach to life-cycle optimisation of vehicle electrification

Although durable goods with low energy consumption are being promoted to achieve a decarbonised society, from the perspective of life-cycle assessment, the choice of new durable goods may increase CO2 emissions. To address this problem, research has been conducted on product replacement based on life-cycle optimisation (LCO), a method for identifying a replacement life span that minimises life-cycle CO2 emissions. However, several additional assumptions complicate the analysis of replacement patterns of products and conditional formulas because cumulative emissions do not increase linearly when considering energy mix and technology improvement, and it is difficult to extend the model to optimisation methods in previous LCO studies. This study developed a new LCO approach by applying the shortest path problem to graph theory. Our methodology can contribute to the following: (i) it is computationally inexpensive; (ii) it is intuitively easy to add complex conditions, such as various policy scenarios and parameter changes; and (iii) once the graph of replacement patterns is defined, the optimal solution can be derived using existing solution methods, such as the Dijkstra algorithm. As a case study, we focused on vehicle replacement, which is a major source of CO2 emissions and is being electrified. In particular, we identified vehicle switching paths that minimise life-cycle CO2 emissions by considering changes in Japan’s energy mix and alternative fuel vehicle (AFV) characteristics. We determined that the optimal vehicle replacement path method to reduce CO2 emissions is to switch first to plug-in hybrid electric vehicles (PHEVs) and then to battery electric vehicles (BEVs). Thus, we suggest that the transition to electric vehicles requires a step-by-step process. This methodology is not only conducive to AFV deployment for decarbonisation but can also be applied to other products, such as air conditioners and lighting. Thus, various transition policies could be formulated using our methodology.

Speeding up the Passenger Assignment Problem in transit planning by the Direct Link Network representation

When designing a public transport network with the passenger travel time in mind, the passenger assignment problem (PAP) needs to be addressed explicitly. It is a complex problem on its own and even the simplest version requires determination of the shortest path for all demand pairs in the network. It is typically treated as a subproblem of larger problems by public transport planners, such as the transit network design problem (TNDP). In the TNDP, bi-level optimization models and/or metaheuristics are used where the PAP is solved when evaluating a line plan. In doing so, the PAP is often addressed by representing the transit network with a so-called ‘Change-and-Go’ (CNG) network with dummy transfer nodes in order to model transfer penalties as part of the passenger travel time. Then, in order to solve the PAP, the all-pairs shortest path problem needs to be solved for this CNG network representation. In this paper, we present a much more efficient network representation, ‘Direct Link Network’ (DLN), where additional edges are added instead of additional nodes. We compare the theoretical complexity of both representations and the computation time required to solve the PAP by using CNG and DLN on the most commonly used benchmark networks and also several real-life networks. The results show that with the DLN representation, the PAP can be solved multiple times faster than with CNG. Consequently, DLN can significantly speed up all TNDP algorithms that solve the PAP multiple times when designing a public transport network.

Stochastic Shortest Path Problem on Borel Space Considering Dead-ends and Undesired Terminal States

Stochastic Shortest Path Problem on Borel Space Considering Dead-ends and Undesired Terminal States

Paths, Proofs, and Perfection: Developing a Human-Interpretable Proof System for Constrained Shortest Paths

People want to rely on optimization algorithms for complex decisions but verifying the optimality of the solutions can then become a valid concern, particularly for critical decisions taken by non-experts in optimization. One example is the shortest-path problem on a network, occurring in many contexts from transportation to logistics to telecommunications. While the standard shortest-path problem is both solvable in polynomial time and certifiable by duality, introducing side constraints makes solving and certifying the solutions much harder. We propose a proof system for constrained shortest-path problems, which gives a set of logical rules to derive new facts about feasible solutions. The key trait of the proposed proof system is that it specifically includes high-level graph concepts within its reasoning steps (such as connectivity or path structure), in contrast to, e.g., using linear combinations of model constraints. Thus, using our proof system, we can provide a step-by-step, human-auditable explanation showing that the path given by an external solver cannot be improved. Additionally, to maximize the advantages of this setup, we propose a proof search procedure that specifically aims to find small proofs of this form using a procedure similar to A* search. We evaluate our proof system on constrained shortest path instances generated from real-world road networks and experimentally show that we may indeed derive more interpretable proofs compared to an integer programming approach, in some cases leading to much smaller proofs.

Efficient Constraint Generation for Stochastic Shortest Path Problems

Current methods for solving Stochastic Shortest Path Problems (SSPs) find states’ costs-to-go by applying Bellman backups, where state-of-the-art methods employ heuristics to select states to back up and prune. A fundamental limitation of these algorithms is their need to compute the cost-to-go for every applicable action during each state backup, leading to unnecessary computation for actions identified as sub-optimal. We present new connections between planning and operations research and, using this framework, we address this issue of unnecessary computation by introducing an efficient version of constraint generation for SSPs. This technique allows algorithms to ignore sub-optimal actions and avoid computing their costs-to-go. We also apply our novel technique to iLAO* resulting in a new algorithm, CG-iLAO*. Our experiments show that CG-iLAO* ignores up to 57% of iLAO*’s actions and it solves problems up to 8x and 3x faster than LRTDP and iLAO*.

A Framework for Data-Driven Explainability in Mathematical Optimization

Advancements in mathematical programming have made it possible to efficiently tackle large-scale real-world problems that were deemed intractable just a few decades ago. However, provably optimal solutions may not be accepted due to the perception of optimization software as a black box. Although well understood by scientists, this lacks easy accessibility for practitioners. Hence, we advocate for introducing the explainability of a solution as another evaluation criterion, next to its objective value, which enables us to find trade-off solutions between these two criteria. Explainability is attained by comparing against (not necessarily optimal) solutions that were implemented in similar situations in the past. Thus, solutions are preferred that exhibit similar features. Although we prove that already in simple cases the explainable model is NP-hard, we characterize relevant polynomially solvable cases such as the explainable shortest path problem. Our numerical experiments on both artificial as well as real-world road networks show the resulting Pareto front. It turns out that the cost of enforcing explainability can be very small.

The study on mechanical model considering optimal self-adaption in the bottleneck area

It aims to solve the problem that the evacuation state of pedestrians depicted by the traditional social force model in a crowded multiexit scenario has a relatively large difference with the actual state, especially the 'optimal path' considered by the self-driving force is the problem of shortest path, and the multiexit evacuation mode depicted by the 'herd behavior' is the local optimum problem. Through in-depth analysis of actual evacuation data of pedestrians and causes of problem, a new crowd evacuation optimization model is established in order to effectively improve the simulation accuracy of crowd evacuation in a multi-exit environment. The model obtains the direction of motion of pedestrians using a field model, fully considers the factors such as exit distance, distribution of pedestrians and regional crowding degree, makes a global optimization for the self-driving force in the social force model using a centralized and distributed network model, and makes a local optimization for it using an elephant herding algorithm, so as to establish a new evacuation optimization method for optimal self-adaption in the bottleneck area. The performance status is compared between the improved social force model and the new model by experiments, and the key factors that affect the new model are analyzed in an in-depth manner. The results show that the new model can optimize the optimal path choice at the early stage of evacuation and improve the evacuation efficiency of pedestrians at the late stage, so as to ensure relatively even distribution of pedestrians at each exit, and also make the simulated evacuation process be more real; and the improvement in overall evacuation efficiency is greater when the number of pedestrians to be evacuated is larger. Therefore, the new model provides a method to solve the phenomenon of disorder in overall pedestrian evacuation due to excessive crowd density during the process of multi-exit evacuation.

A two-stage method for doubly resource constrained elementary shortest path problems

The resource constrained shortest path problem (RCSPP) has been extensively studied. However, its more general version, the doubly resource constrained (elementary) shortest path problem (DRCESPP), is usually not concerned and no exact solution algorithm has been found in the literature. In this work, an exact two-stage method is proposed for solving the DRCESPP. For leveraging resource lower limits, an estimation to the maximum resource consumption of all elementary source–destination paths is proposed in the first stage. This makes the network be reduced more significantly and the algorithm be terminated in the first stage. By introducing an effective search direction into the conventional Pulse algorithm, a LB-first Pulse algorithm is proposed in the second stage to find the exact solution from the reduced network output in the first stage. Extensive computational experiments on various test instances are conducted. The numerical results show that the maximum resource consumption estimation method and the improved Pulse algorithm are effective, and the proposed two-stage method outperforms the Pulse algorithm in solving complex instances.

Machine learning-enhanced optical tweezers for defect-free rearrangement

Optical tweezers constitute pivotal tools in Atomic, Molecular, and Optical (AMO) physics, facilitating precise trapping and manipulation of individual atoms and molecules. This process affords the capability to generate desired geometries in both one-dimensional and two-dimensional spaces, while also enabling real-time reconfiguration of atoms. Due to stochastic defects in these tweezers, which cause catastrophic performance degradation especially in quantum computations, it is essential to rearrange the tweezers quickly and accurately. Our study introduces a machine learning approach that uses the Proximal Policy Optimization model to optimize this rearrangement process. This method focuses on efficiently solving the shortest path problem, ensuring the formation of defect-free tweezer arrays. By implementing machine learning, we can calculate optimal motion paths under various conditions, resulting in promising results in model learning. This advancement presents new opportunities in tweezer array rearrangement, potentially boosting the efficiency and precision of quantum computing research.

SRNN-RSA: a new method to solving time-dependent shortest path problems based on structural recurrent neural network and ripple spreading algorithm

AbstractInfluenced by external factors, the speed of vehicles in the traffic network is changing all the time, which makes the traditional static shortest route unable to meet the real logistics distribution needs. Considering that the existing research on time-dependent shortest path problems (TDSPP) do not include the topological information of the traffic network, it is unable to reflect the spatial and temporal dynamic characteristics of the traffic network during the vehicle travelling process and is unable to update to the changes of the vehicle speed in real time, and poor scalability. Therefore, we used the structural RNN (SRNN) model containing topological information of the road network is used to predict time-varying speeds in the traffic road network. We proposed an SRNN-RSA framework for solving the TDSPP problem, which achieves a synergistic evolution between the real-time vehicle speed change process and the RSA solving process, and the scalability of the proposed SRNN-RSA is demonstrated and validated using different real data. Compared with other algorithms, the results show that SRNN-RSA has the lowest error with the actual situation, which can balance the solution accuracy and calculation speed and is more consistent with the real traffic road network, with better stability and expandability.

A Reliability-Based Network Equilibrium Model with Electric Vehicles and Gasoline Vehicles

With the popularity of electric vehicles, they have become an indispensable part of traffic flow on the road network. This paper presents a reliability-based network equilibrium model to realise the traffic flow pattern prediction on the road network with electric vehicles and gasoline vehicles, which incorporates travel time reliability, electric vehicles’ driving range and recharge requirement. The mathematical expression of reliable path travel time is derived, and the reliability-based network equilibrium model is formulated as a variational inequality problem. Then a multi-criterion labelling algorithm is proposed to solve the reliable shortest path problem, and a column-generation-based method of the successive average algorithm is proposed to solve the reliability-based network equilibrium model. The applicability and efficiency of the proposed model and algorithm are verified on the Nguyen-Dupuis network and the real road network of Sioux Falls City. The proposed model and algorithm can be extended to other road networks and help traffic managers analyse traffic conditions and make sustainable traffic policies.

Exact Solution of the Single-Picker Routing Problem with Scattered Storage

We present a new modeling approach for the single-picker routing problem with scattered storage (SPRP-SS) for order picking in warehouses. The SPRP-SS assumes that an article is, in general, stored at more than one pick position. The task is then the simultaneous selection of pick positions for requested articles and the determination of a minimum-length picker tour collecting the articles. It is a classical result of Ratliff and Rosenthal that, for given pick positions, an optimal picker tour is a shortest path in the state space of a dynamic program with a linear number of states and transitions. We extend the state space of Ratliff and Rosenthal to include scattered storage so that every feasible picker tour is still a path. The additional requirement to make consistent decisions regarding articles to collect is not modeled in the state space so that dynamic programming can no longer be used for the solution. Instead, demand covering can be modeled as additional constraints in shortest-path problems. Therefore, we solve the resulting model with a mixed-integer (linear) programming (MIP) solver. The paper shows that this approach is not only convenient and elegant for single-block parallel-aisle warehouse SPRP-SSs but also generalizable: the solution principle can be applied to different warehouse layouts (we present additional results for a two-block parallel-aisle warehouse) and can incorporate further extensions. Computational experiments with other approaches for the SPRP-SS show that the new modeling approach outperforms the available exact algorithms regarding computational speed. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms—Discrete. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2023.0075 .

A branch-and-price-and-cut algorithm for the unmanned aerial vehicle delivery with battery swapping

In the UAV-based urban delivery setting, the battery capacity serves as one of the key factors affecting the delivery range of drones. For a typical multi-rotor unmanned aerial vehicle (UAV), the maximum effective working range of a lithium battery for a single flight leg is approximately 20 km. In practice, the lithium battery is replaced each time the UAV completes a delivery, even though the remaining power of the replaced lithium battery is still capable of supporting the following flight legs. Therefore, the one-order-one-swapping strategy significantly brings down the battery utilisation as well as the efficiency of the delivery system. In the present paper, the UAV-based delivery problem with battery swapping decision is considered, where multiple en route pickup and deliveries are allowed. To obtain optimal decisions in terms of flight route, time and location of each battery swapping, a branch-and-price-and-cut algorithm is introduced. First we formulate this problem as a mixed integer linear programming (MILP) model, and then decompose it into a master problem and a pricing sub-problem, where the master problem is considered as a set covering model, and the sub-problem as an elementary shortest path problem with resource constraints (ESPPRC). To speed up the process of column generation, efficient heuristics are also integrated to provide basic feasible solutions. In the pricing sub-problem, a combination of heuristic algorithm and dynamic programming is implemented to improve the computational efficiency. In the numerical experiments, the proposed algorithm is tested on various scales of UAV-based urban delivery instances, the performance of which is analysed and compared with the off-the-shelf commercial solver.