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Consider a situation in which individuals –the buyers –have different valuations for the products of a given set. An envy-free assignment of product items to buyers requires that the items obtained by every buyer be purchased at a price not larger than his/her valuation, and each buyer's welfare (difference between product value and price) be the largest possible. Under this condition, the problem of finding prices maximizing the seller's revenue is known to be APX -hard even for unit-demand bidders (with several other inapproximability results for different variants), that is, when each buyer wishes to buy at most one item. Here, we focus on Envy-free Complete Allocation , the special case where a fixed number of copies of each product is available, each of the n buyers must get exactly one product item, and all the products must be sold. This case is known to be solvable in O (n 4) time. We revisit a series of results on this problem and, answering a question found in Leonard (1983), show how to solve it in O (n 3) time by connections to perfect matchings and shortest paths. [ABSTRACT FROM AUTHOR]

Abstract Shortest Distance Path problem has been studied extensively in the literature. This is an important problem with a wide range of applications in the world particularly, transportation, trip planning, etc. Various mathematical models have been proposed in the literature to solve the basic problem and its variations. In this study, we incorporate weather forecast into trip planning. Due to adverse weather conditions, shortest distance path to follow may vary. Furthermore, travel times also become less predictable. Various weather forecast scenarios are generated following a given distribution. Furthermore, stochastic travel times are also considered as part of the analysis. In this study, GA will produce multiple acceptable solutions where fitness function values for each acceptable solution can vary. The average value of arriving time will be considered as the fitness function for a chromosome after simulating the chromosome under different conditions randomly 100 times. [ABSTRACT FROM AUTHOR]

The constantly increasing number of vehicles and the immense size and complexity of road maps set a tough scenario for real world routing. In spite of the tremendous efforts done up to date to tackle this problem, finding the shortest-path in practice is still a challenge due to memory and time constrains. Moreover, most of the efforts have been focused on algorithmics, setting aside the management of the data. However, memory and time constraints are very important to actually construct real world applications, advising a new global approach. In this study we propose a holistic strategy for finding the shortest-path based on efficiently managing the road map data. Our proposal is based on the tile map partitioning, a logic geographical partition strategy. We have developed a routing system highly scalable based on a micro steady state evolutionary algorithm to find the optimal tile map partitioning. We show the actual efficiency and scalability by using the road maps of Malaga, Spain, and Mexico City, making it clear the significant reductions in the time needed to compute the shortest-path (in a real application), what is a key issue that can be freely exploited in future open software for maps. [ABSTRACT FROM AUTHOR]

Abstract We study a pathfinding problem of directed trees under the following situation: each edge has a nonnegative integer length, but the length is unknown in advance and should be searched by a procedure whose computational cost becomes very large as the length increases. We present a method that gives a shortest path. Under some assumptions, the proposed method is optimal in the number of procedure calls for the longest edge length. This graph problem is motivated by procedure synthesis for system configuration in cloud systems. [ABSTRACT FROM AUTHOR]

The choroid is vascular tissue located underneath the retina and supplies oxygen to the outer retina; any damage to this tissue can be a precursor to retinal diseases. This paper presents an automated method of choroidal segmentation from Enhanced Depth Imaging Optical Coherence Tomography (EDI-OCT) images. The Dijkstra shortest path algorithm is used to segment the choroid–sclera interface (CSI), the outermost border of the choroid. A novel intensity-normalisation technique that is based on the depth of the choroid is used to equalise the intensity of all non-vessel pixels in the choroid region. The outer boundary of choroidal vessel and CSI are determined approximately and incorporated to the edge weight of the CSI segmentation to choose optimal edge weights. This method is tested on 190 B-scans of 10 subjects against choroid thickness (CTh) results produced manually by two graders. For comparison, results obtained by two state-of-the-art automated methods and our proposed method are compared against the manual grading, and our proposed method performed the best. The mean root-mean-square error (RMSE) for finding the CSI boundary by our method is 7.71 ± 6.29 pixels, which is significantly lower than the RMSE for the two other state-of-the-art methods ( 36.17 ± 11.97 pixels and 44.19 ± 19.51 pixels). The correlation coefficient for our method is 0.76 , and 0.51 and 0.66 for the other two state-of-the-art methods. The interclass correlation coefficients are 0.72, 0.43 and 0.56 respectively. Our method is highly accurate, robust, reliable and consistent. This identification can enable to quantify the biomarkers of the choroidin large scale study for assessing, monitoring disease progression as well as early detection of retinal diseases. Identification of the boundary can help to determine the loss or change of choroid, which can be used as features for the automatic determination of the stages of retinal diseases. [ABSTRACT FROM AUTHOR]

We introduce Shortest Connection Game , a two-player game played on a directed graph with edge costs. Given two designated vertices in which the players start, the players take turns in choosing edges emanating from the vertex they are currently located at. This way, each of the players forms a path that origins from its respective starting vertex. The game ends as soon as the two paths meet, i.e., a connection between the players is established. Each player has to carry the cost of its chosen edges and thus aims at minimizing its own total cost. In this work we analyse the computational complexity of Shortest Connection Game . On the negative side, Shortest Connection Game turns out to be computationally hard even on restricted graph classes such as bipartite, acyclic and cactus graphs. On the positive side, we can give a polynomial time algorithm for cactus graphs when the game is restricted to simple paths. [ABSTRACT FROM AUTHOR]

The sewer network optimal hydraulic design, for a specific layout consisting of a series of pipes, is the combination of diameters and slopes along the series that satisfy all the hydraulic, commercial, and construction constraints, while minimizing the construction costs. This work explains an exact, exhaustive, and flexible framework to find the cost-optimal design of a series of sewer pipes using a DP-based optimization engine [1] . The hydraulic design problem is modeled as a Shortest Path Problem [2] , where the underlying graph considers every feasible combination of diameter and slope for each pipe in the series. As a result, a shortest path on the graph encodes the optimal hydraulic design decision for the series of pipes. The proposed methodology ensures the global optimal solution from an economic point of view because the graph considers all possible alternatives and the Bellman-Ford [3] algorithm implicitly explores all of them. Without having to simplify hydraulic constraints, this methodology still obtains the optimal solution in a very short computational time using a standard desktop computer. To evaluate the performance of the methodology, several numerical examples are presented varying the pipe material, the topography, and the number of pipes in the series. [ABSTRACT FROM AUTHOR]

In this work we address a game theoretic variant of the shortest path problem, in which two decision makers (players) move together along the edges of a graph from a given starting vertex to a given destination. The two players take turns in deciding in each vertex which edge to traverse next. The decider in each vertex also has to pay the cost of the chosen edge. We want to determine the path where each player minimizes its costs taking into account that also the other player acts in a selfish and rational way. Such a solution is a subgame perfect equilibrium and can be determined by backward induction in the game tree of the associated finite game in extensive form. We show that the decision problem associated with such a path is PSPACE-complete even for bipartite graphs both for the directed and the undirected version. The latter result is a surprising deviation from the complexity status of the closely related game Geography . On the other hand, we can give polynomial time algorithms for directed acyclic graphs and for cactus graphs even in the undirected case. The latter is based on a decomposition of the graph into components and their resolution by a number of fairly involved dynamic programming arrays. Finally, we give some arguments about closing the gap of the complexity status for graphs of bounded treewidth. [ABSTRACT FROM AUTHOR]

• An auxiliary algorithm is proposed to construct initial stage of the MCDM methods. • A multi-objective fuzzy optimization model is formulated. • An extensive comparative analysis is conducted in terms of the addressed methods. • The stability and validity of the results are tested and discussed. • Key findings and managerial insights are provided. This paper examines the performances of the multi-criteria decision-making (MCDM) methods and optimization model in solving multi-attribute shortest path problems such as the safest shortest path under a fuzzy environment. To the best of the knowledge of the authors, this is the first study performing comparative analysis on finding the multi-attribute shortest path by employing well-known techniques in terms of computational effort and results in a fuzzy environment. To this end, the safest shortest path problem, where the risk and distance values concerning arcs on a directed network are defined as triangular fuzzy numbers, is handled. The solution process is carried out under two main headings: (i) To start the solution with MCDM methods, an auxiliary algorithm that constructs a fuzzy decision matrix is proposed. Then, Fuzzy-Technique for Order Preference by Similarity to Ideal Solution (F-TOPSIS), Fuzzy Simple Additive Weighting (F-SAW), and Fuzzy Evaluation Based on Distance from Average Solution (F-EDAS), that are fuzzy-based MCDM methods, are employed to rank the alternative paths. (ii) A multi-objective fuzzy optimization model is formulated, and the most reasonable paths are obtained considering different α-cut levels. Following that, comparative analysis is performed through a set of scenarios considering the different weights of the criteria to see the variability in the rankings. Besides, the addressed fuzzy-based MCDM methods are compared in terms of computational complexity. Overall, the main findings and managerial insights regarding the effectiveness and performance of the methods discussed in the solution process are provided. [ABSTRACT FROM AUTHOR]

The Knödel graph W Δ , n is a graph of even order n and degree Δ where 2 ≤ Δ ≤ ⌊ log 2 ⁡ n ⌋ . Most properties of the Knödel graph are known only for W Δ , 2 Δ and W Δ − 1 , 2 Δ − 2 . In this paper we study the shortest path problem in W Δ , n for all Δ and n . We give a tight bound on the distance between any two vertices in W Δ , n . We show that for almost all Δ , the presented bound differs from actual distance by at most 2. The proofs are constructive and allow us to present an O ( log ⁡ n ) time algorithm to construct a short path between any pair of vertices in W Δ , n . The diameter of W Δ , n is known only for n = 2 Δ . Using our results on the shortest path problem, we present tight upper and lower bounds on the diameter of the Knödel graph for all Δ and n . [ABSTRACT FROM AUTHOR]

Recently the Hopfield Neural Network (HNN) is employed as an optimization tool to solve shortest path problem in communication networks. The hardware implementation of digital Hopfield neural network is an important issue that is considered in this paper. An efficient systolic architecture is proposed for efficient implementing of digital Hopfield neural networks for solving shortest path problem on Field-Programmable-Gate-Array (FPGA) chips. The VHDL hardware description language is employed for hardware modeling of proposed systolic architecture. The results achieved from simulation and hardware synthesizing demonstrates that the proposed systolic architecture has superior performance than relevant architectures in literature for chip area utilization, convergence and maximum operating frequency. [ABSTRACT FROM AUTHOR]

As shortest path (SP) problem has been one of the most fundamental network optimization problems for a long time, technologies for this problem are still being studied. In this paper, a new method by integrating a path finding mathematical model, inspired by Physarum polycephalum, with extracted one heuristic rule to solve SP problem has been proposed, which is called Rapid Physarum Algorithm (RPA). Simulation experiments have been carried out on three different network topologies with varying number of nodes. It is noted that the proposed RPA can find the optimal path as the path finding model does for most networks. What is more, experimental results show that the performance of RPA surpasses the path finding model on both iterations and solution time. [ABSTRACT FROM AUTHOR]

Abstract: A supply chain network design needs to consider the future probability of reconfiguration due to some problems of disaster or price changes. The objective of this article is to design a reconfigurable supply chain network by optimizing inventory allocation and transportation routing. A two-stage programming is composed according to Benders decomposition by allocating inventory in advance and anticipating the changes of transportation routings; thus the transportation routing is stochastic in nature. In addition, the fuzzy shortest path is developed to solve the problem complexity in terms of the multi-criteria of lead time and capacity with an efficient computational method. The results and analysis indicate that the proposed two-stage programming with fuzzy shortest path surpasses the performance of shortest path problem with time windows and capacity constraint (SPPTWCC) in terms of less computational time and CPU memory consumption. Finally, management decision-making is discussed among other concluding remarks. [Copyright &y& Elsevier]

Abstract: metric spaces constitute a far-reaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a metric space are connected by a unique shortest path . In this paper, we present an efficient algorithm for answering two-point distance queries in rectangular complexes and two of theirs subclasses, ramified rectilinear polygons ( rectangular complexes in which the links of all vertices are bipartite graphs) and squaregraphs ( rectangular complexes arising from plane quadrangulations in which all inner vertices have degrees ⩾4). Namely, we show that for a rectangular complex with n vertices, one can construct a data structure of size so that, given any two points , the shortest path between x and y can be computed in time, where p and q are vertices of two faces of containing the points x and y, respectively, such that and is the distance between p and q in the underlying graph of . If is a ramified rectilinear polygon, then one can construct a data structure of optimal size and answer two-point shortest path queries in time, where Δ is the maximal degree of a vertex of . Finally, if is a squaregraph, then one can construct a data structure of size and answer two-point shortest path queries in time. [Copyright &y& Elsevier]

Abstract: A special and important network structured linear programming problem is the shortest path problem. Classical shortest path problems assume that there are unit of shipping cost or profit along an arc. In many real occasions, various attributes (various costs and profits) are usually considered in a shortest path problem. Because of the frequent occurrence of such network structured problems, there is a need to develop an efficient procedure for handling these problems. This paper studies the shortest path problem in the case that multiple attributes are considered along the arcs. The concept of relative efficiency is defined for each path from initial node to final node. Then, an efficient path with the maximum efficiency is determined. [Copyright &y& Elsevier]

Abstract: A common algorithm to solve the shortest path problem (SPP) is the Dijkstra algorithm. In this paper, a generalized Dijkstra algorithm is proposed to handle SPP in an uncertain environment. Two key issues need to be addressed in SPP with fuzzy parameters. One is how to determine the addition of two edges. The other is how to compare the distance between two different paths with their edge lengths represented by fuzzy numbers. To solve these problems, the graded mean integration representation of fuzzy numbers is adopted to improve the classical Dijkstra algorithm. A numerical example of a transportation network is used to illustrate the efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: This paper investigates an oriented spanning tree (OST) based genetic algorithm (GA) for the multi-criteria shortest path problem (MSPP) as well as the multi-criteria constrained shortest path problem (MCSPP). By encoding a path as an OST, in contrast with the existing evolutionary algorithms (EA) for shortest path problem (SPP), the designed GA provides a “search from a paths set to another paths set” mutation mechanism such that both of its local search and global search capabilities are greatly improved. Because the possibility to find a feasible path in a paths set is usually larger than that of only one path is feasible, the designed GA has more predominance for solving MCSPPs. Some computational tests are presented and the test results are compared with those obtained by a recent EA of which the encoding approach and the ideas of evolution operators such as mutation and crossover are adopted in most of the existing EAs for shortest path problems. The test results indicate that the new algorithm is available for both of MSPP and MCSPP. [Copyright &y& Elsevier]

Abstract: Uncertainty theory provides a new tool to deal with the shortest path problem with nondeterministic arc lengths. With help from the operational law of uncertainty theory, this paper gives the uncertainty distribution of the shortest path length. Also, it investigates solutions to the -shortest path and the most shortest path in an uncertain network. It points out that there exists an equivalence relation between the -shortest path in an uncertain network and the shortest path in a corresponding deterministic network, which leads to an effective algorithm to find the -shortest path and the most shortest path. Roughly speaking, this algorithm can be broken down into two parts: constructing a deterministic network and then invoking the Dijkstra algorithm. [Copyright &y& Elsevier]

Abstract: Multi-objective shortest path problem (MOSP) is an extension of a traditional single objective shortest path problem that seeks for the efficient paths satisfying several conflicting objectives between two nodes of a network. MOSP is one of the most important problems in network optimization with wide applications in telecommunication industries, transportation and project management. This research presents an algorithm based on multi-objective ant colony optimization (ACO) to solve the bi-objective shortest path problem. To analyze the efficiency of the algorithm and check for the quality of solutions, experimental analyses are conducted. Two sets of small and large sized problems that generated randomly are solved. Results on the set problems are compared with those of label correcting solutions that is the most known efficient algorithm for solving MOSP. To compare the Pareto optimal frontiers produced by the suggested ACO algorithm and the label correcting algorithm, some performance measures are employed that consider and compare the distance, uniformity distribution and extension of the Pareto frontiers. The results on the set of instance problems show that the suggested algorithm produces good quality non-dominated solutions and time saving in computation of large-scale bi-objective shortest path problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose an efficient method for implementing Dijkstra''s algorithm for the Single Source Shortest Path Problem (SSSPP) in a graph whose edges have positive length, and where there are few distinct edge lengths. The SSSPP is one of the most widely studied problems in theoretical computer science and operations research. On a graph with n vertices, m edges and K distinct edge lengths, our algorithm runs in time if , and time, otherwise. We tested our algorithm against some of the fastest algorithms for SSSPP on graphs with arbitrary but positive lengths. Our experiments on graphs with few edge lengths confirmed our theoretical results, as the proposed algorithm consistently dominated the other SSSPP algorithms, which did not exploit the special structure of having few distinct edge lengths. [Copyright &y& Elsevier]

• Definition of the Shared Resource-Constrained Multi-Shortest Path Problem, SRMSPP. • A parallelizable matheuristic to solve the SRMSPP. • Instances with thousands of networks and millions of arcs are solved. • The SRMSPP can be applied to some Scheduling Problems. We tackle the problem of finding, for each network within a collection, the shortest path between two given nodes, while not exceeding the limits of a set of shared resources. We present an integer programming (IP) formulation of this problem and propose a parallelizable matheuristic consisting of three phases: (1) generation of feasible solutions, (2) combination of solutions, and (3) solution improvement. We show that the shortest paths found with our procedure correspond to the solution of some type of scheduling problems such as the Air Traffic Flow Management (ATFM) problem. Our computational results include finding optimal solutions to small and medium-size ATFM instances by applying Gurobi to the IP formulation. We use those solutions to assess the quality of the output produced by our proposed matheuristic. For the largest instances, which correspond to actual flight plans in ATFM, exact methods fail and we assess the quality of our solutions by means of Lagrangian bounds. Computational results suggest that the proposed procedure is an effective approach to the family of shortest path problems that we discuss here. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents the investigations on the application of particle swarm optimization (PSO) to solve shortest path (SP) routing problems. A modified priority-based encoding incorporating a heuristic operator for reducing the possibility of loop-formation in the path construction process is proposed for particle representation in PSO. Simulation experiments have been carried out on different network topologies for networks consisting of 15–70 nodes. It is noted that the proposed PSO-based approach can find the optimal path with good success rates and also can find closer sub-optimal paths with high certainty for all the tested networks. It is observed that the performance of the proposed algorithm surpasses those of recently reported genetic algorithm based approaches for this problem. [Copyright &y& Elsevier]

Abstract: Shortest path finding has a variety of applications in transportation and communication.In this paper, we study a well-known self-stabilizing algorithm for the shortest path problem for the distributed systems. The previous works on this topic had two assumptions that can be relaxed in this paper. First, in the previous works, the systems were assumed to be integral-weighted, whereas in this paper, the systems are real-weighted. Second, and more importantly, the previous works have shown that the algorithm is self-stabilizing under the more restricted central demon model, whereas in this paper, we give a rigorous proof showing that the algorithm is actually self-stabilizing under the more general distributed demon model. The work in this paper is of significance because in the existing literature on self-stabilizing systems, most of the papers regarding the distributed demon are for the ring networks only; there are very few papers that discuss the self-stabilizing algorithms for the general distributed systems assuming the distributed demon model. [Copyright &y& Elsevier]

• Stratified sampling reduces genotype diversify thus reducing exploration time. • Opposition-Based guessing increases initial population fitness. • Path Length diversity is more important during the initialization phase. • The repair function can introduce diversity to population. Initialization is the first and a major step in the implementation of evolutionary algorithms (EAs). Although there are many common general methods to initialize EAs such as the pseudo-random number generator (PRNG), there is no single method that can fit every problem. This study provides a new, flexible, diversity-aware, and easy-to-implement initialization method for a genetic algorithm for the shortest path problem. The proposed algorithm, called stratified opposition-based sampling (SOBS), considers phenotype and genotype diversity while striving to achieve the best fitness for the initialization population. SOBS does not depend on a specific type of sampling, because the main goal is to stratify the sampling space. SOBS aims at an initial population with higher fitness and diversity in the phenotype and genotype. To investigate the performance of SOBS, four network models were used to simulate real-world networks. Compared with the most frequently used initialization method, that is, PRNG, SOBS provides more accurate solutions, better running time with less memory usage, and an initial population with higher fitness. Statistical analysis showed that SOBS yields solutions with higher accuracy in 68–100% of the time. Although this study was focused on the genetic algorithm, it can be applied to other population-based EAs that solve the shortest path problem and use the same direct population representation such as particle swarm optimization (PSO). [ABSTRACT FROM AUTHOR]