Constrained Shortest Path Problem Research Articles

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.

Enhanced methods for the weight constrained shortest path problem

AbstractThe classic problem of constrained pathfinding is a well‐studied, yet challenging, network optimization problem with a broad range of applications in various areas such as communication and transportation. The weight constrained shortest path problem (WCSPP), the base form of constrained pathfinding with only one side constraint, aims to plan a cost‐optimum path with limited weight/resource usage. Given the bi‐criteria nature of the problem (i.e., dealing with the cost and weight of paths), methods addressing the WCSPP have some common properties with bi‐objective search. This article leverages the recent state‐of‐the‐art techniques in both constrained pathfinding and bi‐objective search and presents two new solution approaches to the WCSPP on the basis of A* search, both capable of solving hard WCSPP instances on very large graphs. We empirically evaluate the performance of our algorithms on a set of large and realistic problem instances and show their advantages over the state‐of‐the‐art algorithms in both time and space metrics. This article also investigates the importance of priority queues in constrained search with A*. We show with extensive experiments on both realistic and randomized graphs how bucket‐based queues without tie‐breaking can effectively improve the algorithmic performance of exhaustive A*‐based bi‐criteria searches.

PathWyse: a flexible, open-source library for the resource constrained shortest path problem

In this paper, we consider a fundamental and hard combinatorial problem: the Resource Constrained Shortest Path Problem (RCSPP). We describe the implementation of a flexible, open-source library for the solution of the RCSPP, called PathWyse, capable of tackling several variants of the problem. We designed PathWyse with the final user in mind, developing easy-to-use interfaces without compromising performance. We provide computational experiments on three classes of instances of the RCSPP, namely RCSPP on cyclic networks, RCSPP on large acyclic networks, and RCSPP on ad-hoc cyclic networks. We show that PathWyse is packed off-the-shelf with algorithms capable of tackling generic problems, and can exploit dedicated algorithms for specific classes. This paper represents the first step along the journey of devising and implementing a comprehensive open-source library for a large variety of RCSPPs. The current version of the library carries exact algorithms for the RCSPP but new algorithms, both heuristic and exact, will be added thanks to the flexible design. We also foresee PathWyse becoming a platform ready for data-driven and process-driven methodologies for these types of problems.

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%.

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

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

A Directional Heuristics Pulse Algorithm for a Two Resources Constrained Shortest Path Problem with Reinitialization

<p style='text-indent:20px;'>In this paper, the path planning problem of UAV flight error correction is studied, and a mixed-integer linear programming (MILP) model is proposed to solve the resource constrained shortest path problem (RCSPP-R) with reinitialization, and its resource accumulation can be reinitialized to zero at the reset node. Since RCSPP-R is a NP-hard problem, Pulse algorithm is used to solve the proposed model. Aiming at the high computational cost of pulse algorithm, three pruning strategies are proposed to improve the effectiveness and efficiency of the algorithm, namely, preprocessing strategy based on direction constraint to reduce the search space, dynamic relaxation strategy to improve the quality of boundary, and heuristic search strategy to optimize the node expansion sequence. Then the time-consuming of the algorithm and the number of dominant solutions are used as evaluation indexes for comparison. Numerical results show that the proposed algorithm can deal with this problem more reliably. Specifically, compared with the Label Correction algorithm, the original Pulse algorithm and the Ant Colony algorithm, there are advantages in the number of superior solutions, and compared with the above three algorithms, the time consumption is reduced by 31.64%, 88.52% and 83.5% respectively.</p>

一种基于牛顿法的约束最短路问题算法

一种基于牛顿法的约束最短路问题算法

Development of Linear Battery Model for Path Planning with Mixed Integer Linear Programming: Simulated and Experimental Validation

Mixed Integer Linear Programs (MILPs) are often used in the path planning of both ground and aerial vehicles. Such a formulation of the path planning problem requires a linear objective function and constraints, limiting the fidelity of the tracking of vehicle states. One such state often used is the charge level of the on board battery. High-fidelity battery state estimation requires nonlinear differential equations to be solved. This state estimation is vital in path planning to ensure flyable paths, however when using a linear path planning problem cannot implement these nonlinear models. Poor accuracy in battery estimation during the path planning runs the risk of the planned path being feasible by the estimation model but in reality will deplete the battery to a critical level, resulting in a dangerous planned path. To the end of higher accuracy battery estimation within a linear framework, we test a simple linear battery model which predicts the change in state-of-charge (SOC) of a battery given a power draw, time duration, and current SOC in the context of an a-priori path planning problem. This context differentiates itself from real-time estimation. In ahead-of-time path planning, the changes to battery draw are often assumed as a series of constant power draws as opposed to rapidly changing power draw which may occur in real-time battery tracking and estimation. The linear battery model is presented and then tested against alternate models in both numerical and in experimental tests. Further, the effect of the proposed linear model on the time-to-solve a resource constrained shortest path problem is also evaluated, where two different algorithms are used to solve the path planning problem. It is seen that the linear model performs well in battery state estimation while remaining implementable in a Linear Program or MILP, with minimal effect on the time-to-solve. This provides what we consider to be a worthwhile trade-off in improved accuracy relative to increased time-to-solve.

Partial and complete replanning of an intermodal logistic system under disruptions

The exclusive and excessive use of long-distance road transportation is not suitable way to reduce the negative environmental impacts of logistics systems. Intermodal transport, combining road with other transport modes, has the potential to reduce both operating costs and carbon footprints. One of the reasons for the low share of intermodal transport is its requirement for the coordination of scheduled transport services that can result in reducing reliability in case of disruptions due to the arrival of new shipment orders, fluctuations in shipment quantities, delays, and service cancellations within the network. This calls for reliable and efficient algorithms to replan the shipments’ distribution. In this paper, the replanning problem is formulated as a path-based multi-commodity network flows. We provide two different network topologies, one of which is based on a time–space network, while the other embeds time aspect in a highly scalable alternative structure to large transportation networks. We propose a column generation method whose pricing sub-problems are presented as resource constrained shortest path problem solved via a tailored label-correcting algorithm. We look at the pros and cons of complete and partial replanning in case of disruption and provide managerial insights for intermodal networks. An extensive set of computational experiments is presented on realistic instances being generated with the consultation of our industrial partners for a logistic network including railways, waterways, and roads. The promising outcomes validate the efficiency of the proposed approach that can be easily adjusted to real-time intermodal logistic replanning.

Machine-Learning–Based Arc Selection for Constrained Shortest Path Problems in Column Generation

Column generation is an iterative method used to solve a variety of optimization problems. It decomposes the problem into two parts: a master problem and one or more pricing problems (PP). The total computing time taken by the method is divided between these two parts. In routing or scheduling applications, the problems are mostly defined on a network, and the PP is usually an NP-hard shortest path problem with resource constraints. In this work, we propose a new heuristic pricing algorithm based on machine learning. By taking advantage of the data collected during previous executions, the objective is to reduce the size of the network and accelerate the PP, keeping only the arcs that have a high chance to be part of the linear relaxation solution. The method has been applied to two specific problems: the vehicle and crew scheduling problem in public transit and the vehicle routing problem with time windows. Reductions in computational time of up to 40% can be obtained. Funding: The authors thank Giro, Inc., and the Natural Sciences and Engineering Research Council of Canada for financial support [Grant CRDPJ 520349-17].

Exact algorithms based on a constrained shortest path model for robust serial-batch and parallel-batch scheduling problems

We study robust single-machine batch scheduling problems under uncertain processing times to minimize total flow time. Two types of batches are considered: serial batch (s-batch) and parallel batch (p-batch). These problems can model many on-site production and logistics applications which involve uncertain factors such as defect rates. We first prove that a sequencing rule for the shortest nominal processing time is optimal for both s-batch and p-batch problems. We then propose polynomial-time algorithms based on the observation that the robust batch scheduling problems are reducible or partially reducible to a constrained shortest path problem through worst-case scenario analysis. We further present more efficient algorithms for the special case of uniform maximum deviation times for all jobs. The algorithms are evaluated computationally, and the results show that their performance is satisfactory on the tested instances.

Finding a Shortest Non-Zero Path in Group-Labeled Graphs

We study a constrained shortest path problem in group-labeled graphs with nonnegative edge length, called the shortest non-zero path problem. Depending on the group in question, this problem includes two types of tractable variants in undirected graphs: one is the parity-constrained shortest path/cycle problem, and the other is computing a shortest noncontractible cycle in surface-embedded graphs. For the shortest non-zero path problem with respect to finite abelian groups, Kobayashi and Toyooka (2017) proposed a randomized, pseudopolynomial-time algorithm via permanent computation. For a slightly more general class of groups, Yamaguchi (2016) showed a reduction of the problem to the weighted linear matroid parity problem. In particular, some cases are solved in strongly polynomial time via the reduction with the aid of a deterministic, polynomial-time algorithm for the weighted linear matroid parity problem developed by Iwata and Kobayashi (2021), which generalizes a well-known fact that the parity-constrained shortest path problem is solved via weighted matching. In this paper, as the first general solution independent of the group, we present a rather simple, deterministic, and strongly polynomial-time algorithm for the shortest non-zero path problem. This result captures a common tractable feature behind the parity and topological constraints in the shortest path/cycle problem. The algorithm is based on Dijkstra’s algorithm for the unconstrained shortest path problem and Edmonds’ blossom shrinking technique in matching algorithms; this approach is inspired by Derigs’ faster algorithm (1985) for the parity-constrained shortest path problem via a reduction to weighted matching. Furthermore, we improve our algorithm so that it does not require explicit blossom shrinking, and make the computational time match Derigs’ one. In the speeding-up step, a dual linear programming formulation of the equivalent problem based on potential maximization for the unconstrained shortest path problem plays a key role.

Weight Constrained Path Finding with Bidirectional A*

Weight constrained path finding, known as a challenging variant of the classic shortest path problem, aims to plan cost optimum paths whose weight/resource usage is limited by a side constraint. Given the bi-criteria nature of the problem (i.e., the presence of cost and weight), solutions to the Weight Constrained Shortest Path Problem (WCSPP) have some properties in common with bi-objective search. This paper leverages the state-of-the-art bi-objective search algorithm BOBA* and presents WC-BA*, an exact A*-based WCSPP method that explores the search space in different objective orderings bidirectionally. We also enrich WC-BA* with two novel heuristic tuning approaches that can significantly reduce the number of node expansions in the exhaustive search of A*. The results of our experiments on a large set of realistic problem instances show that our new algorithm solves all instances and outperforms the state-of-the-art WCSPP algorithms in various scenarios.

A graph algorithm for the time constrained shortest path

Highly efficient algorithms for solving the time constrained shortest path problem have been highlighted over the past decades to reduce the cost of vehicle travel in the road network. The paper presents a novel graph algorithm comprising three stages to acquire a time constrained shortest path between any two nodes on the map. In the first stage, the undirected graph is transformed into a directed graph (DT graph) by deleting edges that must not be included in any of the time constrained shortest paths. A variant DT tree with the destination as the root and the source as the leaf is then constructed from the DT graph in the second stage. By finding the minimal difference value between leaves and the root of the variant DT tree, we can eventually obtain a time constrained shortest path from the variant DT tree in the third stage. Experimental results show that our algorithm not only requires less running time than some classical graph algorithms but also can work immediately in the first stage without knowing the information of both the destination and a specific time constraint.

Optimization of Charging Strategies for Battery Electric Vehicles Under Uncertainty

The comparably low driving ranges of battery electric vehicles (BEV) cause time-consuming recharging stops if long distances have to be covered. Thus, navigation systems not only have to compute routes leading from the BEV’s current position to the destination, but also to plan recharging stops. This type of routing problem is often modeled as a constrained shortest path problem. The constraint ensures that the BEV does not run out of energy. In this paper, a de facto deterministic reformulation of this problem type is suggested, which allows handling uncertainty–particularly the risks resulting from imperfect energy consumption predictions. For this purpose, a certain part of the battery capacity is used as an energy buffer. Different approaches to dynamically optimize the size of this energy buffer in dependency of the expected level of uncertainty are proposed and a corresponding modification of a typical routing algorithm is described. Furthermore, a simulation study is conducted showing that the described framework allows keeping the probability to run out of energy close to zero (for the test settings: < 0.5%) as long as a suitable approach for defining the size of the energy buffer is applied.

A column generation approach for the driver scheduling problem with staff cars

Given a set of timetabled bus trips, transport companies are faced with the challenge of finding a feasible driver schedule that covers all trips and abides by various labor union regulations. The regulations are primarily concerned with providing sufficient breaks for the drivers during the day. Practical limitations in the city network enforce drivers to travel by cars between bus stops to have breaks. Transport companies have a limited number of cars, known as staff cars, which have to be returned to their respective depots at the end of the day. The simultaneous scheduling of drivers and staff cars for the drivers is known as the driver scheduling problem with staff cars (DSPSC). It is estimated that the DSPSC accounts for 60% of a bus company’s operational expense, and this paper proposes a column generation approach that attempts to minimize operational expense. The column-generation framework iterates between a master problem, a subproblem for generating driver variables and a subproblem for generating staff car variables. The subproblem related to the drivers is formulated as a resource constrained shortest path problem, which is solved by a dynamic programming approach. Several heuristic branching strategies are explored to find integer solutions. The proposed methodology is tested on eight real-life instances from seven Northern European bus companies. A comparison with a state-of-the-art mixed integer programming (MIP) solver and an adaptive large neighborhood search (ALNS) heuristic indicate that the column generation approach provides improved solutions for six instances and the average improvement is 1.45%.

A heuristics pulse algorithm with relaxation pruning strategy for resources re-initialized UAV path planing

Path planning is the basis and prerequisite for unmanned aerial vehicle (UAV) to perform tasks, and it is important to achieve precise location in path planning. This paper focuses on solving the UAV path planning problem under the constraint of system positioning error. Some nodes can re-initiate the accumulated flight error to zero and this type of scenario can be modeled as the resource-constrained shortest path problem with re-initialization (RCSPP-R). The additional re-initiation conditions expand the set of viable paths for the original constrained shortest path problem and increasing the search cost. To solve the problem, an effective preprocessing method is proposed to reduce the network nodes. At the same time, a relaxed pruning strategy is introduced into the traditional Pulse algorithm to reduce the search space and avoid more redundant calculations on unfavorable scalable nodes by the proposed heuristic search strategy. To evaluate the accuracy and effectiveness of the proposed algorithm, some numerical experiments were carried out. The results indicate that the three strategies can reduce the search space by 99%, 97% and 80%, respectively, and in the case of a large network, the heuristic algorithm combining the three strategies can improve the efficiency by an average of 80% compared to some classical solution.

Branch and Price for Bus Driver Scheduling with Complex Break Constraints

This paper presents a Branch and Price approach for a real-life Bus Driver Scheduling problem with a complex set of break constraints. The column generation uses a set partitioning model as master problem and a resource constrained shortest path problem as subproblem. Due to the complex constraints, the branch and price algorithm adopts several novel ideas to improve the column generation in the presence of a high-dimensional subproblem, including exponential arc throttling and a dedicated two-stage dominance algorithm. Evaluation on a publicly available set of benchmark instances shows that the approach provides the first provably optimal solutions for small instances, improving best-known solutions or proving them optimal for 48 out of 50 instances, and yielding an optimality gap of less than 1% for more than half the instances.

A Fast Exact Algorithm for the Resource Constrained Shortest Path Problem

Resource constrained path finding is a well studied topic in AI, with real-world applications in different areas such as transportation and robotics. This paper introduces several heuristics in the resource constrained path finding context that significantly improve the algorithmic performance of the initialisation phase and the core search. We implement our heuristics on top of a bidirectional A* algorithm and evaluate them on a set of large instances. The experimental results show that, for the first time in the context of constrained path finding, our fast and enhanced algorithm can solve all of the benchmark instances to optimality, and compared to the state of the art algorithms, it can improve existing runtimes by up to four orders of magnitude on large-size network graphs.

Fuzzy Constrained Shortest Path Problem for Location-Based Online Services

One of the important issues under discussion connected with traffic on the roads is improving transportation. In this regard, spatial information, including the shortest path, is of particular importance due to the reduction of economic and environmental costs. Here, the constrained shortest path (CSP) problem which has an important application in location-based online services is considered. The aim of this problem is to find a path with the lowest cost where the traversal time of the path does not exceed from a predetermined time bound. Since precise prediction of cost and time of the paths is not possible due to traffic and weather conditions, this paper discusses the CSP problems with fuzzy cost and fuzzy time. After formulating the CSP problem an efficient algorithm for finding the constrained optimal path is designed. The application of the proposed model is presented on a location-based online service called Snap.