Parametric search for the bi-attribute concave shortest path problem

Keyword index volumes 72–79 (1994)

Labeling methods for partially ordered paths

The shortest path problem (SPP) constitutes one of the most frequently encountered classes of problems in graph theory. It is certainly the most fundamental of components in the fields of transportation and communication networks. Shortest path problems may be encountered directly, possibly as a result of a clever formulation of a problem not at first sight involving shortest paths, or indirectly as a subproblem in the solution of a more complicated optimization problem. This use of shortest path problems as subroutines motivates the search for algorithms with good theoretical bounds on running time. We also seek computer implementations whose empirical performances are rapid in spite of perhaps weak theoretical bounds for the algorithms they implement. As testimony to the importance of shortest path and related problems, a large number of surveys, annotated bibliographies, and reviews have appeared over the past thirty years. Among them are those by Dreyfus (1969), Pierce (1975), Golden and Magnanti (1977), and Gallo and Pallottino (1988).

Shortest path problem with uncertain arc lengths

The constrained shortest path (CSP) problem has been widely used in transportationoptimization, crew scheduling, network routing and so on. It is an open issue since it is a NP-hard problem. In this paper, we propose an innovative method which is based on the internal mechanism of the adaptive amoeba algorithm. The proposed method is divided into two parts. In the first part, we employ the original amoeba algorithm to solve the shortest path problem in directed networks. In the second part, we combine the Physarum algorithm with a bio-inspired rule to deal with the CSP. Finally, by comparing the results with other method using an examples in DCLC problem, we demonstrate the accuracy of the proposed method.

This paper presents a particle swarm optimization (PSO) algorithm for solving the delay constrained least (DCLC) path problem, i.e., shortest path problem (SPP) with a delay constraint on the total cost of the optimal path. The proposed algorithm uses the principle of Lagrange relaxation based aggregated cost, where PSO and noising metaheuristic are used for minimizing the modified function. It essentially consists of two PSOs. The main PSO is basically a hybrid PSONoising metaheuristic algorithm for efficient global search for the minimization part of the DCLC-Lagrangian relaxation by finding multiple shortest paths between a source and a destination. The second/auxiliary PSO is used to obtain the optimal Lagrangian multiplier for solving the maximization part of the Lagrangian relaxation of the DCLC path problem. For the main PSO, a new path encoding/decoding scheme based on heuristics has been devised for representing the paths as particles. The comparative simulation results on several networks with random topologies illustrate the efficiency of the proposed hybrid algorithm for constrained shortest path computation.

Solving the shortest path problem is important in achieving high performance or to efficiently utilize resources in various kinds of networks, e.g., data communication networks and transportation networks. Fortunately, under independent additive link weights, this problem is solvable in polynomial time. However, in many real-life networks, the link weights (e.g., delay, bandwidth, failure probability) are often correlated due to spatial or temporal dependencies. These correlated link weights together might behave in a different manner and are not always additive. In this paper, we first propose two correlated link-weight models, namely (i) the deterministic correlated model and (ii) the (log-concave) stochastic correlated model. Subsequently, we study the shortest path problem under these two correlated models. We prove that the shortest path problem is NP-hard under the deterministic correlated model, and even cannot be approximated to arbitrary degree in polynomial time. On the other hand, we show that the shortest path problem is polynomial-time solvable under a nodal deterministic correlated model. Finally, we show that the shortest path problem under the (log-concave) stochastic correlated model can be solved by convex optimization.

We study the 2-Disjoint Shortest Paths (2-DSP) problem: given a directed weighted graph and two terminal pairs (s₁,t₁) and (s₂,t₂), decide whether there exist vertex-disjoint shortest paths between each pair. Building on recent advances in disjoint shortest paths for DAGs and undirected graphs (Akmal et al. 2024), we present an O(mn log n)-time algorithm for this problem in weighted directed graphs that do not contain negative or zero weight cycles. This algorithm presents a significant improvement over the previously known O(m⁵n)-time bound (Berczi et al. 2017). Our approach exploits the algebraic structure of polynomials that enumerate shortest paths between terminal pairs. A key insight is that these polynomials admit a recursive decomposition, enabling efficient evaluation via dynamic programming over fields of characteristic two. Furthermore, we demonstrate how to report the corresponding paths in O(mn² log n)-time. In addition, we extend our techniques to a more general setting: given two terminal pairs (s₁, t₁) and (s₂, t₂) in a directed graph, find the minimum possible number of vertex intersections between any shortest path from s₁ to t₁ and s₂ to t₂. We call this the Minimum 2-Disjoint Shortest Paths (Min-2-DSP) problem. We provide in this paper the first efficient algorithm for this problem, including an O(m² n³)-time algorithm for directed graphs with positive edge weights, and an O(m+n)-time algorithm for DAGs and undirected graphs. Moreover, if the number of intersecting vertices is at least one, we show that it is possible to report the paths in the same O(m+n)-time. This is somewhat surprising, as there is no known o(mn) time algorithm for explicitly reporting the paths if they are vertex-disjoint, and is left as an open problem in (Akmal et al. 2024).

This research addresses how the stochasticity and correlation of travel speeds affect the shortest path solutions in sustainable environments. We consider a shortest path problem with the objective function of minimizing a linear combination of the mean and standard deviation of carbon emissions. By adjusting the proportion of the standard deviation in the objective function, the effects of speed stochasticity and correlation are studied under different preferences of the decision-makers on the fluctuations of carbon emissions. Based on 102-day real speed data from the Los Angeles freeway network, this research conducts extensive numerical experiments on 200 randomly chosen origin-destination pairs. Experimental results demonstrate the necessity of considering speed stochasticity and correlation, especially when the standard deviation of carbon emissions takes a large proportion in the objective function. As the weight of the standard deviation in the objective function increases from 0 to 1.5, the reduction of emission objective values increases from 0.03% to 0.13% by considering speed stochasticity, and increases from 0.02% to 0.20% by considering speed correlation. Taking the city Los Angeles with about 2361 taxis and about 525,945 passenger orders in January 2017 as an example, 0.03% and 0.02% reductions respond to about 3156 kg and 2630 kg carbon emission, respectively.

Finding the minimum cost in the shortest path (SP) problem is one of the most important and fundamental problems in network applications. In the current chapter due to the uncertainty caused by the data that occurs in the real world, we use various tools in the fuzzy set theory to perform the cost of arc lengths. First, assume that the arcs are in the form of interval fuzzy sets. Using the well-known max-min criterion provided by Bellman and Zadeh for fuzzy decision making, model formulation is performed as a problem of nonlinear and mixed number programming. then describe the algorithm based on reliability, thereafter another method based on interval-valued calculation and Dijkstra’s algorithm mentioned. at the end with example, we analyze the performance of the methods and compare the results. Then we introduce the (SP) problem in hesitant fuzzy environment and develop the two mentioned methods to solve hesitant fuzzy shortest path (HFSP) problem. Numerical example shows the efficiency of developed methods. Then we show the length of the arcs with hesitant fuzzy sets. In this case, we solve the fuzzy shortest path problem by introducing a new distance function and ranking. Numerical results are obtained using LINGO and MATLAB software.

A distributed algorithm is presented that can be used to solve the single-destination shortest path (SDSP) problem or the all-pairs shortest path (APSP) problem for a class of clustered data networks. The network graph is assumed to be characterized with a balanced hierarchically clustered (BHC) topology. The BHC topology is introduced in this paper and is shown to be a realistic characterization for a large class of interconnected data networks. For certain types of BHC topologies, the SDSP problem can be solved with computation and communication time complexities of O(log n), assuming one processor is available at each of the n number of nodes. Assuming p processors are available at each node, computation and communication time complexities of O((n/p) log n) and O(n log n) are achievable, respectively, for solving the APSP problem. It is also shown that the algorithm converges in an asynchronous environment. >

The finding of shortest paths in networks is one of the most basic combinatorial optimization problems, and the algorithms for solving these problems are among the most widely used. The reader is probably aware that a variety of optimization problems, some of which appear to have little to do with paths in networks, can be formulated as shortest path problems.

The shortest path problem is fundamental in graph theory and has been studied extensively due to its practical importance. Despite this aspect, finding the shortest path between two nodes remains a significant challenge in many applications, as it often becomes complex and time consuming. This complexity becomes even more challenging when constraints make the problem non-additive, thereby increasing the difficulty of finding the optimal path. The objective of this paper is to present a broad perspective on the conventional shortest path problem. It introduces a new method to classify cost functions associated with graphs by defining distinct sets of cost functions. This classification facilitates the exploration of line graphs and an understanding of the upper bounds on the transformation sizes for these types of graphs. Based on these foundations, the paper proposes a practical methodology for solving non-additive shortest path problems. It also provides a proof of optimality and establishes an upper bound on the algorithmic cost of the proposed methodology. This study not only expands the scope of traditional shortest path problems but also highlights their computational complexity and potential solutions.

In the article, the definition of an undirected multiple graph of any natural multiplicity $$k > 1$$ is stated. There are edges of three types: ordinary edges, multiple edges, and multi-edges. Each edge of the last two types is the union of $$k$$ linked edges, which connect 2 or $$k + 1$$ vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common ending vertex to $$k$$ linked edges of some multi-edge. If a vertex is the common end of a multi-edge, it cannot be the common end of any other multi-edge. Also, a class of the divisible multiple graphs is considered. The main peculiarity of them is a possibility to divide the graph into $$k$$ parts, which are adjusted on the linked edges and which have no common ordinary edges. Each part is an ordinary graph. The following terms are generalized: the degree of a vertex, connectedness of a graph, the path, the cycle, the weight of an edge, and the path length. The definition of a reachability set for the ordinary and multiple edges is stated. The adjacency property is defined for a pair of reachability sets. It is shown, that we can check the connectedness of a multiple graph with the polynomial algorithm based on the search for reachability sets and testing their adjacency. A criterion of the existence of a multiple path between two given vertices is considered. The shortest multiple path problem is stated. Then, we suggest an algorithm for finding the shortest path in a multiple graph. It uses Dijkstra’s algorithm for finding the shortest paths in subgraphs, which correspond to different reachability sets.

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