A fuzzy shortest path with the highest reliability

This paper concentrates on a shortest path problem on a network where arc lengths (costs) are not deterministic numbers, but imprecise ones. Here, costs of the shortest path problem are fuzzy intervals with increasing membership functions, whereas the membership function of the total cost of the shortest path is a fuzzy interval with a decreasing linear membership function. By the max–min criterion suggested in [R.E. Bellman, L.A. Zade, Decision-making in a fuzzy environment, Management Science 17B (1970) 141–164], the fuzzy shortest path problem can be treated as a mixed integer nonlinear programming problem. We show that this problem can be simplified into a bi-level programming problem that is very solvable. Here, we propose an efficient algorithm, based on the parametric shortest path problem for solving the bi-level programming problem. An illustrative example is given to demonstrate our proposed algorithm.     

A Dijkstra-like shortest path algorithm for certain cases of negative arc lengths

If each negative length arc of a digraphG is acyclic, i.e., does not belong to any cycle, then we show that the shortest paths from a given node to all other nodes can be computed inO(V ²) time, whereV is the number of nodes inG.     

Solving the shortest path tour problem

In this paper, we study the shortest path tour problem in which a shortest path from a given origin node to a given destination node must be found in a directed graph with non-negative arc lengths. Such path needs to cross a sequence of node subsets that are given in a fixed order. The subsets are disjoint and may be different-sized. A polynomial-time reduction of the problem to a classical shortest path problem over a modified digraph is described and two solution methods based on the above reduction and dynamic programming, respectively, are proposed and compared with the state-of-the-art solving procedure. The proposed methods are tested on existing datasets for this problem and on a large class of new benchmark instances. The computational experience shows that both the proposed methods exhibit a consistent improved performance in terms of computational time with respect to the existing solution method.     

Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints

When vehicle routing problems with additional constraints, such as capacity or time windows, are solved via column generation and branch-and-price, it is common that the pricing subproblem requires the computation of a minimum cost constrained path on a graph with costs on the arcs and prizes on the vertices. A common solution technique for this problem is dynamic programming. In this paper we illustrate how the basic dynamic programming algorithm can be improved by bounded bi-directional search and we experimentally evaluate the effectiveness of the enhancement proposed. We consider as benchmark problems the elementary shortest path problems arising as pricing subproblems in branch-and-price algorithms for the capacitated vehicle routing problem, the vehicle routing problem with distribution and collection and the capacitated vehicle routing problem with time windows.     

New efficient shortest path simplex algorithm: pseudo permanent labels instead of permanent labels

We introduce a new network simplex pivot rule for the shortest path simplex algorithm. This new pivot rule chooses a subset of non-basic arcs to simultaneously enter into the basis. We call this operation a multiple pivot. We show that a shortest path simplex algorithm with this pivot rule performs O(n) multiple pivots and runs in O(nm) time. Our pivot rule is based on the new concept of a pseudo permanently labeled node, and it can be adapted to design a new label-correcting algorithm that runs in O(nm). Moreover, this concept lets us introduce new rules to identify negative cycles. Finally, we compare the network simplex algorithm with multiple pivots with other previously proposed efficient network simplex algorithm in a computational experiment.     

The robust shortest path problem in series-parallel multidigraphs with interval data

In this paper the robust shortest path problem in edge series-parallel multidigraphs with interval costs is examined. The maximal regret criterion is applied to calculate the optimal solution. It is shown that this problem is NP-hard. A pseudopolynomial algorithm for the studied problem is constructed.     

LP formulations of the shortest path tree problem

Based on a pair of primal-dual LP formulations of the shortest path tree problem, the first algorithmic approach to reoptimizing the shortest paths subject to changes in the edge weights was proposed by S. Pallottino and M.G. Scutellá in 2003. We shall here focus solely on their introductory sections, propose some simplifications of the models considered, and finally relate the resulting models to the presentation of single-source shortest path problems in textbooks treating this subject with but limited or no reference to LP.Received: April 2004, Revised: August 2004, MSC classification: 90C05, 90C35, 90B10 Dedicated to the memory of Stefano Pallottino     

An aggregate label setting policy for the multi-objective shortest path problem

We consider label setting algorithms for the multi-objective shortest path problem with any number of sum and bottleneck objectives. We propose a weighted sum aggregate ordering of the labels, specifically tailored to combine sum and bottleneck objectives. We show that the aggregate order leads to a consistent reduction of solution times (up to two-thirds) with respect to the classical lexicographic order.     

A note on chance constrained programming with fuzzy coefficients

This paper deals with nonlinear chance constrained programming as well as multiobjective case and goal programming with fuzzy coefficients occurring in not only constraints but also objectives. We also present a fuzzy simulation technique for handling fuzzy objective constraints and fuzzy goal constraints. Finally, a fuzzy simulation based genetic algorithm is employed to solve a numerical example.     

An applicable method for solving the shortest path problems

A theorem of Hardy, Littlewood, and Polya, first time is used to find the variational form of the well known shortest path problem, and as a consequence of that theorem, one can find the shortest path problem via quadratic programming. In this paper, we use measure theory to solve this problem. The shortest path problem can be written as an optimal control problem. Then the resulting distributed control problem is expressed in measure theoretical form, in fact an infinite dimensional linear programming problem. The optimal measure representing the shortest path problem is approximated by the solution of a finite dimensional linear programming problem.     

The robust shortest path problem with interval data via Benders decomposition

Many real problems can be modelled as robust shortest path problems on digraphs with interval costs, where intervals represent uncertainty about real costs and a robust path is not too far from the shortest path for each possible configuration of the arc costs. In this paper we discuss the application of a Benders decomposition approach to this problem. Computational results confirm the efficiency of the new algorithm. It is able to clearly outperform state-of-the-art algorithms on many classes of networks. For the remaining classes we identify the most promising algorithm among the others, depending of the characteristics of the networks. Received: September 2004 / Accepted: March 2005 AMS classification: 90C47, 52B05, 90C57 The work was partially supported by the European Commission IST projects MOSCA (IST-2000-29557) and BISON (IST-2001-38923). All correspondence to: Roberto Montemanni     

Accelerated label setting algorithms for the elementary resource constrained shortest path problem

A label setting algorithm for solving the Elementary Resource Constrained Shortest Path Problem, using node resources to forbid repetition of nodes on the path, is implemented. A state-space augmenting approach for accelerating run times is considered. Several augmentation strategies are suggested and compared numerically.     

Two disjoint shortest paths problem with non-negative edge length

In the two disjoint shortest paths problem ( 2-DSPP), the input is a graph (or a digraph) and its vertex pairs $(s_1,t_1)$ and $(s_2,t_2)$, and the objective is to find two vertex-disjoint paths $P_1$ and $P_2$ such that $P_i$ is a shortest path from $s_i$ to $t_i$ for $i=1,2$, if they exist. In this paper, we give a first polynomial-time algorithm for the undirected version of the 2-DSPP with an arbitrary non-negative edge length function.     

A branch and bound algorithm for the robust shortest path problem with interval data

Many real problems can be modelled as robust shortest path problems on interval digraphs, where intervals represent uncertainty about real costs and a robust path is not too far from the shortest path for each possible configuration of the arc costs.A branch and bound algorithm for this problem is presented.     

Time version of the shortest path problem in a stochastic-flow network

Many studies on hardware framework and routing policy are devoted to reducing the transmission time for a flow network. A time version of the shortest path problem thus arises to find a quickest path, which sends a given amount of data from the unique source to the unique sink with minimum transmission time. More specifically, the capacity of each arc in the flow network is assumed to be deterministic. However, in many real-life networks, such as computer systems, telecommunication systems, etc., the capacity of each arc should be stochastic due to failure, maintenance, etc. Such a network is named a stochastic-flow network. Hence, the minimum transmission time is not a fixed number. We extend the quickest path problem to evaluating the probability that d$d$ units of data can be sent under the time constraint T$T$. Such a probability is named the system reliability. In particular, the data are transmitted through two minimal paths simultaneously in order to reduce the transmission time. A simple algorithm is proposed to generate all (d,T)$(d,T)$-MPs and the system reliability can then be computed in terms of (d,T)$(d,T)$-MPs. Moreover, the optimal pair of minimal paths with highest system reliability could be obtained.     

An extension of labeling techniques for finding shortest path trees

Label setting techniques are all based on Dijkstra’s condition of always scanning the node with the minimum label, which guarantees that each node will be scanned exactly once; while this condition is sufficient it is not necessary. In this paper, we discuss less restrictive conditions that allow the scanning of a node that does not have the minimum label, yet still maintaining sufficiency in scanning each node exactly once; various potential shortest path schemes are discussed, based on these conditions. Two approaches, a label setting and a flexible hybrid one are designed and implemented. The performance of the algorithms is assessed both theoretically and computationally. For comparative analysis purposes, three additional shortest path algorithms – the commonly cited in the literature – are coded and tested. The results indicate that the approaches that rely on the less restrictive optimality conditions perform substantially better for a wide range of network topologies.     

A parametric total order on fuzzy numbers and a fuzzy shortest route problem

A parametric total order relation is introduced in the form of a certain modification of the "fuzzy max" order on the class of fuzzy numbers generated by a shape function. By the parametric relation. fuzzy numbers unordered with respect of the fuzzy max order can be ordered according either to their value of center or the size of ambiguity. A fuzzy shortest route problem in which are distances are given by fuzzy numbers is discussed under the criterion of the parametric total order, and solved by the dynamic programming approach. A method is proposed to find all of fuzzy routes mimmal in the sense of the fuzzy max order.     

Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs

This paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network. For both problems, we determine the maximum and minimum weights that each edge can have so that a given path remains optimal. For both problems, we show how to determine these maximum and minimum values for all edges in O(m + K log K) time, where m is the number of edges in the network, and K is the number of edges on the given optimal path.     

Classification of reconfiguration graphs of shortest path graphs with no induced 4-cycles

For any graph $G$ with $a,b\in V\left(G\right)$, a shortest path reconfiguration graph can be formed with respect to $a$ and $b$; we denote such a graph as $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths from $a$ to $b$ in $G$ while two vertices $U,W$ in $V\left(S(G,a,b)\right)$ are adjacent if and only if the vertex sets of the paths that represent $U$ and $W$ differ in exactly one vertex. In a recent paper (Asplund et al., 2018), it was shown that shortest path graphs with girth five or greater are exactly disjoint unions of even cycles and paths. In this paper, we extend this result by classifying all shortest path graphs with no induced 4-cycles.