Nonnegative Arc Lengths Research Articles

An efficient exact approach for the constrained shortest path tour problem

ABSTRACTGiven a directed graph with non-negative arc lengths, the Constrained Shortest Path Tour Problem () is aimed at finding a shortest path from a single-origin to a single-destination, such that a sequence of disjoint and possibly different-sized node subsets are crossed in a given fixed order. Moreover, the optimal path must not include repeated arcs. In this paper, for the we propose a new mathematical model and a new efficient Branch & Bound method. Extensive computational experiments have been carried out on a significant set of test problems in order to evaluate empirically the performance of the proposed approach.

Robust shortest path planning and semicontractive dynamic programming

AbstractIn this article, we consider shortest path problems in a directed graph where the transitions between nodes are subject to uncertainty. We use a minimax formulation, where the objective is to guarantee that a special destination state is reached with a minimum cost path under the worst possible instance of the uncertainty. Problems of this type arise, among others, in planning and pursuit‐evasion contexts, and in model predictive control. Our analysis makes use of the recently developed theory of abstract semicontractive dynamic programming models. We investigate questions of existence and uniqueness of solution of the optimality equation, existence of optimal paths, and the validity of various algorithms patterned after the classical methods of value and policy iteration, as well as a Dijkstra‐like algorithm for problems with nonnegative arc lengths.© 2016 Wiley Periodicals, Inc. Naval Research Logistics 66:15–37, 2019

The constrained shortest path tour problem

In this paper, we study the constrained shortest path tour problem. Given a directed graph with non-negative arc lengths, the aim is to find a single-origin single-destination shortest path, which needs to cross a sequence of node subsets that are given in a fixed order. The subsets are disjoint and may be of different size. In addition, it is required that the path does not include repeated arcs.Theoretical properties of the problem are studied, proving that it belongs to the complexity class NP-complete. To exactly solve it, a Branch & Bound method is proposed. Given the problem hardness, a Greedy Randomized Adaptive Search Procedure is also developed to find near-optimal solutions for medium to large scale instances.Extensive computational experiments, on a significant set of test problems, are carried out in order to empirically evaluate the performance of the proposed approaches. The computational results show that the Greedy Randomized Adaptive Search Procedure is effective in finding optimal or near optimal solutions in very limited computational time.

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.

Complexity analysis and optimization of the shortest path tour problem

The shortest path tour problem (SPTP) consists in finding a shortest path from a given origination node s to a given destination node d in a directed graph with nonnegative arc lengths with the constraint that the optimal path P should successively and sequentially pass through at least one node from given node subsets T1, T2, . . . , TN, where \({T_i \cap T_j = \emptyset, \forall\ i, j=1,\ldots,N,\ i \neq j}\). In this paper, it will proved that the SPTP belongs to the complexity class P and several alternative techniques will be presented to solve it.

New polynomial-time cycle-canceling algorithms for minimum-cost flows

The cycle-canceling algorithm is one of the earliest algorithms to solve the minimum-cost flow problem. This algorithm maintains a feasible solution x in the network G and proceeds by augmenting flows along negative-cost directed cycles in the residual network G(x) and thereby canceling them. For the minimum-cost flow problem with integral data, the generic version of the cycle-canceling algorithm runs in pseudopolynomial time, but several polynomial-time specific implementations can be obtained by specifying the choices of cycles to be canceled. In this paper, we describe a new polynomial-time implementation of the cycle-canceling algorithm. Our algorithm is a scaling algorithm and proceeds by augmenting flows along negative cycles with “sufficiently large” residual capacity. Further, it identifies such a cycle by solving a shortest path problem with nonnegative arc lengths. For a network with n nodes and m arcs, our cycle-canceling algorithm performs O(m log(nU)) augmentations and runs in O(m(m + n log n) log (nU)) time, where U is an upper bound on the node supplies/demands and finite arc capacities. We also show that the cycle-canceling algorithm (i) can solve the uncapacitated minimum-cost flow problem in O(n(m + n log n) log (nU)) time; (ii) can obtain an integer optimal solution of the convex cost-flow problem in O(m(m + n log n) log (nU)) time; and (iii) can be modified so that it runs in O(m(m + n log n) min {log (nU), m log n}) time, which is a strongly polynomial time bound. © 2000 John Wiley & Sons, Inc.

An efficient implementation of an algorithm for findingK shortest simple paths

In this article, we present an efficient computational implementation of an algorithm for finding the K shortest simple paths connecting a pair of vertices in an undirected graph with n vertices, m arcs, and nonnegative arc lengths. A minimal number of intermediate paths is formed based on the method of Katoh, Ibaraki and Mine [Networks 12 (1982), 411–427], which has the lowest worst-case complexity of O(n2) among all other existing algorithms for this problem. A theoretical description of the algorithm is presented with detailed explanations and some examples of the more complicated steps. Efficient data structures for storing and retrieving a large number of paths are given. The results of wide-ranging experimentation with a large number of randomly generated graphs of varying size and density confirm the linear dependency of computing time on K, as proven in Katoh et al., and the quadratic dependency of computing time on graph size as suggested by the worst-case computational complexity. © 1999 John Wiley & Sons, Inc. Networks 34: 88–101, 1999

On the Robust Shortest Path Problem

The shortest path problem is of great importance to the real world in such areas as transportation, network design, telecommunication, etc. The deterministic version of the problem is easily solved. However, in the real world, uncertainty is frequently encountered and must be dealt with. For instance, in a transportation network, the path that has the least travel time when no traffic is present might be prone to accidents and traffic jams, and thus it may lead to a drastic increase in travel time during rush hours. In situations with significant uncertainty, the deterministic approach can be far from sufficient. New and appropriate criteria and models to handle uncertainties along with efficient solution techniques are in need. In this paper, we take a minimax approach to measure the performance under uncertainties. Under our measure, one path is superior to all others only if its worst case performance is better than that of the others. The complexity of the defined models, exact and heuristic solution methods are explored and presented. Analysis is done to examine the performance of the heuristic method. The shortest path (SP) problem in a network with nonnegative arc lengths can be solved easily by Dijkstra's labeling algorithm in polynomial time. In the case of significant uncertainty of the arc lengths, a robustness approach is more appropriate. In this paper, we study the SP problem under arc length uncertainties. A scenario approach is adopted to characterize uncertainties. Two robustness criteria are specified: the absolute robust criterion and the robust deviation criterion. We show that under both criteria the robust SP problem is NP-complete even for the much more restricted layered networks of width 2, and with only 2 scenarios. A pseudo-polynomial algorithm is devised to solve the robust SP problem in general networks under bounded number of scenarios. Also presented is a more efficient algorithm for layered networks. However, in the case of unlimited number of scenarios, we show that the robust SP problem is strongly NP-hard. A simple heuristic for finding a good robust shortest path is provided, and the worst case performance is analyzed.

A capacity scaling algorithm for the constrained maximum flow problem

AbstractThe constrained maximum flow problem is to send the maximum possible flow from a source node s to a sink node t in a directed network subject to a budget constraint that the cost of flow is no more than D. In this paper, we consider two versions of this problem: (i) when the cost of flow on each arc is a linear function of the amount of flow, and (ii) when the cost of flow is a convex function of the amount of flow. We suggest capacity scaling algorithms that solve both versions of the constrained maximum flow problem in O((m log M) S(n, m)) time, where n is the number of nodes in the network; m, the number of arcs; M, an upper bound on the largest element in the data: and S(n, m), the time required to solve a shortest path problem with nonnegative arc lengths. Our algorithms are generalizations of the capacity scaling algorithms for the minimum cost flow and convex cost flow problems and illustrate the power of capacity scaling algorithms to solve variants of the minimum cost flow problem in polynomial time.

New Polynomial Shortest Path Algorithms and Their Computational Attributes

This paper presents six new variants of the polynomially bounded Partitioning Shortest Path (PSP) algorithm for finding the shortest path from one node to all other nodes in a network. Three of these variants, one for negative arc lengths, but without negative cycles, and two for nonnegative arc lengths, augment the PSP algorithm to maintain a property called sharp by Shier and Witzgall. The other three variants augment the PSP algorithm to maintain a property called near-sharp for nonnegative arc lengths. Extensive computational testing is presented on one of the sharp variants for nonnegative arc lengths and two of the near-sharp variants. The empirical results based on 4500 test problems with 90 different configurations and three different network topologies indicate that these new algorithms have excellent computational performance characteristics. Based on total solution times for the 4500 test problems, these new algorithms out-perform all other algorithms tested. In addition, one of the near-sharp algorithms strictly dominates all others on all problem topologies tested.

最短路算法の実際的評価と新バケット法

For the problem of finding the shortest paths from a prescribed vertex to the other vertices in a given network with nonnegative arc lengths, we investigate practical efficiencies of typical existing algorithms, i.e., the label-correcting method with a FIFO and that of a two-way sequence, and the label-setting method with a heap and that with a 1- or 2-level bucket system. We propose also a new method with a variable bucket system and compare it with those existing methods. Among the existing methods, the label-correcting method of a two-way sequence and the label-setting method with a I-level bucket system have been found efficient in most cases. The new proposed method is not only as efficient as those but it is also robust for a large variety of networks, so that it may be recommended for practical use.

On finding shortest paths in nonnegative networks

A modification of Dantzig's algorithm for the all pairs shortest paths problem is given. The new algorithm applies only to graphs with nonnegative arc lengths. For an N-node complete graph it has a worst case running time of 2 3 N 3 triple operations of the form D ij : = min( D ij , D ik + D kj ) and N 2 log N other comparisons. This contrasts with a lower bound of N( N − 1) ( N − 2) triples in any pure triple operation algorithm, and seems to be the first algorithm in which no operation need be repeated N 3 times. Sparsity and some other conditions may also be utilized.