Shortest path problem with uncertain arc lengths

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).

Parametric search for the bi-attribute concave shortest path problem

Keyword index volumes 72–79 (1994)

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.

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).

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

This paper analyzes the shortest path problem (SPP) in social networks, based on the investigation and implementation of different methods on a simulated example. The objectives of the paper include identification of the most commonly used methods for finding the shortest path in a social network as a strategic attempt to speed the search of network nodes, focusing on the application of the two most used SPP methods: the Dijkstra and Bellman-Ford algorithms. A comparative analysis is used as an investigation method for performance evaluation of different algorithms, based on their implementation and behavior, tested on a social network example. The research results indicate that the Dijkstra algorithm is faster, and therefore more suitable for searching the shortest connection in social networks.

The fuzzy shortest path length and the corresponding shortest path in a network

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.

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.

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.

The reserved delivery service can help the information service providers to provide more consistent performance to their customers by the provisioning of reserved bandwidth on a delivery subnetwork. However, the configuration problem of a reserved delivery subnetwork is a hard optimization problem with no efficient exact algorithm besides exhaust searches. In this paper, we introduce a reserved delivery subnetwork configuration algorithm based on an idea of the maximum sharing shortest path tree (MSSPT). The proposed algorithm is motivated by the observation that the path sharing of multiple flows reduce the cost in reserved delivery subnetworks. Thus, a solution close to the optimal could occur in a subnetwork with the maximum degree of flow sharing. The maximum sharing shortest path tree problem can be categorized as a multicriteria shortest path problem. Using an algorithm based on the shortest path network (SPN, a unique subnetwork in which every path s → u is a shortest path in the original graph), we develop an efficient algorithm for the maximum sharing shortest path problem. The proposed algorithm is an approximation algorithm in nature because it takes the MSSPT as the approximation solution to the reserved delivery subnetwork configuration problem. Our experimental results show that the proposed algorithm has good performance against an easily computed lower bound, but has time complexity comparable to a single source shortest path algorithm.

A class of cooperative games arising from shortest path problems is defined. These shortest path games are totally balanced and allow a population-monotonic allocation scheme. Possible methods for obtaining core elements are indicated; first, by relating to the allocation rules in taxation and bankruptcy problems, second, by constructing an explicit rule that takes opportunity costs into account by considering the costs of the second best alternative and that rewards players who are crucial to the construction of the shortest path. The core and the bargaining sets of Davis-Maschler and Mas-Colell are shown to coincide. Finally, noncooperative games arising from shortest path problems are introduced, in which players make bids or claims on paths. The core allocations of the cooperative shortest path game coincide with the payoff vectors in the strong Nash equilibria of the associated noncooperative shortest path game.