An efficient label setting/correcting shortest path algorithm

We design a new label shortest path algorithm by applying the concept of a pseudo permanent label. This approach allows an algorithm to partition the set of nodes into two new sets: pseudo permanently labeled nodes and its complementary set. From this point of view, this new label method can be considered as a label setting method. Moreover, at least one node becomes permanently labeled when some nodes which belong to the set of pseudo permanently labeled nodes are scanned in each iteration of the algorithm. In the case of networks with non-negative length arcs it is easy to prove that this node has the minimum distance label among the non-pseudo permanently labeled nodes. On the other hand, it is not known during the computation which pseudo permanently labeled nodes are permanently labeled. Therefore, all distance labels are temporary and the algorithm becomes a label correcting method. Nevertheless, the proposed algorithm exhibits some nice features, such as: (1) the time bound for the running of the algorithm for a network with n nodes and m arcs is O(nm); (2) the number of node scan operations in the algorithm is less than the number of these operations in the previous label correcting algorithms as is observed in the computational experience; (3) the algorithm incorporates two new rules which allow easy detection of a negative cycle in the network; (4) the algorithm is quite simple and very easy to implement, and does not require sophisticated data structures; (5) the algorithm exhibits flexibility in the order in which the new pseudo permanently labeled nodes are scanned. The above features are possible through the application of the pseudo permanent label concept.     

Fast and fine quickest path algorithm

We address the quickest path problem proposing a new algorithm based on the fact that its optimal solution corresponds to a supported non-dominated point in the objective space of the minsum–maxmin bicriteria path problem. This result allows us to design a label setting algorithm which improves all existing algorithms in the state-of-the-art, as it is shown in the extensive experiments carried out considering synthetic and real networks.     

On the Hardness of Equal Shortest Path Routing

In telecommunication networks packets are carried from a source s to a destination t on a path that is determined by the underlying routing protocol. Most routing protocols belong to the class of shortest path routing protocols. In such protocols, the network operator assigns a length to each link. A packet going from s to t follows a shortest path according to these lengths. For better protection and efficiency, one wishes to use multiple (shortest) paths between two nodes. Therefore the routing protocol must determine how the traffic from s to t is distributed among the shortest paths. In the protocol called OSPF-ECMP (for Open Shortest Path First-Equal Cost Multiple Path) the traffic incoming at every node is uniformly balanced on all outgoing links that are on shortest paths. In that context, the operator task is to determine the “best” link lengths, toward a goal such as maximizing the network throughput for given link capacities.In this work, we show that the problem of maximizing even a single commodity flow for the OSPF-ECMP protocol cannot be approximated within any constant factor ratio. Besides this main theorem, we derive some positive results which include polynomial-time approximations and an exponential-time exact algorithm. We also prove that despite their weakness, our approximation and exact algorithms are, in a sense, the best possible.     

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.     

Link distance and shortest path problems in the plane

This paper describes algorithms to compute Voronoi diagrams, shortest path maps, the Hausdorff distance, and the Fréchet distance in the plane with polygonal obstacles. The underlying distance measures for these algorithms are either shortest path distances or link distances. The link distance between a pair of points is the minimum number of edges needed to connect the two points with a polygonal path that avoids a set of obstacles. The motivation for minimizing the number of edges on a path comes from robotic motions and wireless communications because turns are more difficult in these settings than straight movements.Link-based Voronoi diagrams are different from traditional Voronoi diagrams because a query point in the interior of a Voronoi face can have multiple nearest sites. Our site-based Voronoi diagram ensures that all points in a face have the same set of nearest sites. Our distance-based Voronoi diagram ensures that all points in a face have the same distance to a nearest site.The shortest path maps in this paper support queries from any source point on a fixed line segment. This is a middle-ground approach because traditional shortest path maps typically support queries from either a fixed point or from all possible points in the plane.The Hausdorff distance and Fréchet distance are fundamental similarity metrics for shape matching. This paper shows how to compute new variations of these metrics using shortest paths or link-based paths that avoid polygonal obstacles in the plane.     

Label Correcting Methods to Solve Multicriteria Shortest Path Problems

In this paper, we deal with the solution of the multicriteria shortest path problem. In particular, we present a class of labeling methods to generate the entire set of Pareto-optimal path-length vectors from an origin node s to all other nodes in a multicriteria network. The proposed methods are supported theoretically by the principle of optimality and they are defined on the basis of various innovative node and label selection strategies.Computational results comparing the proposed methods to state-of-the-art approaches to solve the problem considered are also reported. They indicate that our methods are competitive in general; in several cases, they outperform all the other codes.     

Min–max and min–max regret versions of combinatorial optimization problems: A survey

Min–max and min–max regret criteria are commonly used to define robust solutions. After motivating the use of these criteria, we present general results. Then, we survey complexity results for the min–max and min–max regret versions of some combinatorial optimization problems: shortest path, spanning tree, assignment, min cut, min s–t cut, knapsack. Since most of these problems are NP-hard, we also investigate the approximability of these problems. Furthermore, we present algorithms to solve these problems to optimality.     

Finding K shortest looping paths with waiting time in a time–window network

A time-constrained shortest path problem is a shortest path problem including time constraints that are commonly modeled by the form of time windows. Finding K shortest paths are suitable for the problem associated with constraints that are difficult to define or optimize simultaneously. Depending on the types of constraints, these K paths are generally classified into either simple paths or looping paths. In the presence of time–window constraints, waiting time occurs but is largely ignored. Given a network with such constraints, the contribution of this paper is to develop a polynomial time algorithm that finds the first K shortest looping paths including waiting time. The time complexity of the algorithm is O(rK²|V₁|³), where r is the number of different windows of a node and |V₁| is the number of nodes in the original network.     

Approximation of min–max and min–max regret versions of some combinatorial optimization problems

This paper investigates, for the first time in the literature, the approximation of min–max (regret) versions of classical problems like shortest path, minimum spanning tree, and knapsack. For a constant number of scenarios, we establish fully polynomial-time approximation schemes for the min–max versions of these problems, using relationships between multi-objective and min–max optimization. Using dynamic programming and classical trimming techniques, we construct a fully polynomial-time approximation scheme for min–max regret shortest path. We also establish a fully polynomial-time approximation scheme for min–max regret spanning tree and prove that min–max regret knapsack is not at all approximable. For a non-constant number of scenarios, in which case min–max and min–max regret versions of polynomial-time solvable problems usually become strongly NP-hard, non-approximability results are provided for min–max (regret) versions of shortest path and spanning tree.     

The first <Emphasis Type="Italic">K</Emphasis> minimum cost paths in a time-schedule network

The time-constrained shortest path problem is an important generalisation of the classical shortest path problem and in recent years has attracted much research interest. We consider a time-schedule network, where every node in the network has a list of pre-specified departure times and departure from a node may take place only at one of these departure times. The objective of this paper is to find the first K minimum cost simple paths subject to a total time constraint. An efficient polynomial time algorithm is developed. It is also demonstrated that the algorithm can be modified for finding the first K paths for all possible values of total time.     

Development of the tree-based link labeling algorithm for optimal path-finding in urban transportation networks

The optimal path-finding algorithm which is an important module in developing route guidance systems and traffic control systems has to provide correct paths to consider U-turns, P-turns, and no-left-turns in urban transportation networks.Traditional methods which have been used to consider those regulations on urban transportation networks can be categorized into network representation and algorithmic methods like the vine-building algorithm. First, network representation methods use traditional optimal path-finding algorithms with modifications to the network structure: for example, just adding dummy nodes and links to the existing network allows constraint-search in the network. This method which creates large networks is hard to implement and introduces considerable difficulties in network coding. With the increased number of nodes and links, the memory requirement tremendously increases, which causes the processing speed to slow down. For these reasons, the method has not been widely accepted for incorporating turning regulations in optimal path-finding problems in transportation networks. Second, algorithmic methods, as they are mainly based on the vine-building algorithm, have been suggested for determining optimal path for networks with turn penalties and prohibitions. However, the algorithms, although they nicely reflect the characteristics of urban transportation networks, frequently provide infeasible or suboptimal solutions.The algorithm to be suggested in this research is a method which is basically based on Dijkstra's algorithm [1] and the tree-building algorithm used to construct optimal paths. Unlike the traditional node labeling algorithms which label each node with minimum estimated cost, this algorithm labels each link with minimum estimated cost.Comparison with the vine-building algorithm shows that the solution of the link-labeling algorithm is better than that of the vine-building algorithm which very frequently provides suboptimal solutions. As a result, the algorithm allows turning regulations, while providing an optimal solution within a reasonable time limit.     

On the second point-to-point undirected shortest simple path problem

We address the determination of the second point-to-point shortest simple path in undirected networks. The effective reduced cost concept is introduced to compute the second best solution. This concept is used to prove that a path tree containing the second point-to-point shortest simple path is adjacent to any shortest path tree. Therefore, this result immediately implies a method requiring O(m) time once that the shortest path tree is obtained on an undirected network with n nodes and m edges.     

A survey of algorithms for the generalized assignment problem

This paper surveys algorithms for the well-known problem of finding the minimum cost assignment of jobs to agents so that each job is assigned exactly once and agents are not overloaded. All approaches seem to be based on branch-and-bound with bound supplied through heuristics and through relaxations of the primal problem formulation. From the survey one can select building blocks for the design of one's own tailor-made algorithm. The survey also reveals that although just about every mathematical programming technique was tried on this problem, there is still a lack of a representative set of test problems on which competing enumeration algorithms can be compared, as well as a shortage of effective heuristics.     

Paths, trees and matchings under disjunctive constraints

We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints.We prove that the minimum spanning tree problem is strongly NP-hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP-hard for conflict graphs where every connected component is a single edge.Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.     

Shortest path problem with forbidden paths: The elementary version

This paper addresses the elementary shortest path problem with forbidden paths. The main aim is to find the shortest paths from a single origin node to every other node of a directed graph, such that the solution does not contain any path belonging to a given set (i.e., the forbidden set). It is imposed that no cycle can be included in the solution. The problem at hand is formulated as a particular instance of the shortest path problem with resource constraints and two different solution approaches are defined and implemented. One is a Branch & Bound based algorithm, the other is a dynamic programming approach. Different versions of the proposed solution strategies are developed and tested on a large set of test problems.     

A Biologically Inspired Optimization Algorithm for Solving Fuzzy Shortest Path Problems with Mixed Fuzzy Arc Lengths

The shortest path problem is among fundamental problems of network optimization. Majority of the optimization algorithms assume that weights of data graph’s edges are pre-determined real numbers. However, in real-world situations, the parameters (costs, capacities, demands, time) are not well defined. The fuzzy set has been widely used as it is very flexible and cost less time when compared with the stochastic approaches. We design a bio-inspired algorithm for computing a shortest path in a network with various types of fuzzy arc lengths by defining a distance function for fuzzy edge weights using \(\alpha \) cuts. We illustrate effectiveness and adaptability of the proposed method with numerical examples, and compare our algorithm with existing approaches.     

An exact algorithm for minimum CDS with shortest path constraint in wireless networks

In this paper, we study a minimum connected dominating set problem (CDS) in wireless networks, which selects a minimum CDS with property that all intermediate nodes inside every pairwise shortest path should be included. Such a minimum CDS (we name this problem as SPCDS) is an important tache of some other algorithms for constructing a minimum CDS. We prove that finding such a minimum SPCDS can be achieved in polynomial time and design an exact algorithm with time complexity O(δ ² n), where δ is the maximum node degree in communication graph.     

Stable routing under the Spanning Tree Protocol

The Spanning Tree Protocol routes traffic on shortest path trees. If some edges fail, the traffic has to be rerouted consequently, setting up alternative trees. In this paper we design efficient algorithms to compute polynomial-size integer weights so as to enforce the following stability property: if q=O(1) edges fail, traffic demands that are not affected by the failures are not redirected. Stability is a goal pursued by network operators in order to minimize transmission delays due to the restoration process.     

Efficient Parallel Algorithms for the Minimum Cost Flow Problem

In this paper, we propose efficient parallel implementations of the auction/sequential shortest path and the -relaxation algorithms for solving the linear minimum cost flow problem. In the parallel auction algorithm, several augmenting paths can be found simultaneously, each of them starting from a different node with positive surplus. Convergence results of an asynchronous version of the algorithm are also given. For the -relaxation method, there exist already parallel versions implemented on CM-5 and CM-2; our implementation is the first on a shared memory multiprocessor. We have obtained significant speedup values for the algorithms considered; it turns out that our implementations are effective and efficient.     

An extended shortest path problem: A data envelopment analysis approach

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.