Showing papers in "Networks in 1974"

Disjoint paths in a network

Abstract: Routes between two given nodes of a network are called diversified if they are node-disjoint, except at the terminals. Diversified routes are required for reliability in communication, and an additional criterion is that their total cost, assumed to be the sum of individual arc lengths or costs, is minimum. An algorithm and related theory is described for a general number K of node-disjoint paths with minimum total length. The algorithm applies shortest path labeling algorithms familiar in the literature. K node-disjoint paths are found in K iterations of a single shortest path algorithm.

Networks and vehicle routing for municipal waste collection

Abstract: Vehicle routing for municipal waste collection encompasses a variety of problems. In this paper, we explore the techniques we have developed for solving some of these problems.

A fundamental problem in vehicle routing

Abstract: An important but difficult combinatorial problem, in general, is to find the optimal route for a single vehicle on a given network. This paper defines a problem type, called the General Routing Problem, and gives an algorithm for its solution. The classical Traveling Salesman Problem and the Chinese Postman Problem are shown to be special limiting cases of the General Routing Problem. The algorithm provides a unified approach to both node and arc oriented routing problems, and exploits special properties of most real transportation networks such as sparsity of the associated adjacency matrix, and the tendency for arc symmetry at many nodes. For node oriented routing problems, this approach tends to produce large reduction in effective problem size.

Algorithms on circular‐arc graphs

Abstract: Consider a finite family of non-empty sets. The intersection graph of this family is obtained by representing each set by a vertex, two vertices being connected by an edge if and only if the corresponding sets intersect. The intersection graph of a family of arcs on a circularly ordered set is called a circular-arc graph. In this paper we give a characterization of the circular-arc graph and we describe efficient algorithms for recognizing two subclasses. Also, we describe efficient algorithms for finding a maximum independent set, a minimum covering by cliques and a maximum clique of a circular-arc graph.

Implementation and computational comparisons of primal, dual and primal-dual computer codes for minimum cost network flow problems

Abstract: : The paper presents extensive computational experience with a special purpose primal simplex algorithm. The performance is compared to that of several 'state of the art' out-of-kilter computer codes. The computational characteristics of several different primal feasible start procedures and pivot selection strategies are also examined. The study discloses the advantages, in both computation time and memory requirements, of the primal approach over the out-of-kilter method. The test environment has the following distinguishing properties: (1) all of the codes are tested on the same machine and the same problems, (2) the test set includes capacitated and uncapacitated transhipment networks, transportation problems, and assignment problems, and (3) problem sizes ranging from 100 to 8,000 nodes with up to 35,000 arcs are examined. (Author)

Exact calculation of computer network reliability

Abstract: In designing a distributed computer network, the reliability and availability of the communication paths between all pairs of centers is a primary consideration. Many different approaches have been taken to calculate exactly the probability of successful communication between any specified pair of centers, for a given failure probability of the individual computer systems and communication facilities. However, almost all of these methods are not computationally feasible for large networks. Consequently, approximate calculations of network reliability have been suggested. In this paper a procedure is given for the exact calculation of the probability that all paths between a pair of nodes in a given network are interrupted. The procedure generates mutually exclusive sets of cutting states and calculates the probability of the related events. Summing these values one obtains the probability of service interruption between the specified pair of nodes. If all links fail with equal probability p, the coefficients of a polynomial can be calculated, describing the service disruption probability as a function of p.

Routing a fleet of M vehicles to/from a central facility

Abstract: In a previous paper, algorithms for optimally routing a single vehicle were presented. In this paper, the problem of routing a fleet of M vehicles serving fixed demand for transportation from many origins to a single destination is considered. The M vehicle General Routing Problem (abbreviated M-GRP) is the core of many real routing problems, such as school bus routing for a single school (many-to-one) and garbage truck routing to a central dump. It is shown that the M-GRP on a given graph is equivalent to a GRP on a modified graph. From this result, it is shown that single vehicle routing algorithms can be applied to the M vehicle routing problem.

Computing capacitated minimal spanning trees efficiently

Abstract: An analysis is made of the computational complexity of a class of heuristic algorithms for the solution of the minimal spanning tree problem subject to a restriction on the maximum number (or weight) of nodes in any subtree rooted at a distinguished node. This is of particular interest in designing networks with branch capacity restrictions. The algorithm is a modification of Kruskal's Algorithm where weights are assigned to the nodes and then used, along with the arc lengths, to select the order in which arcs are considered for inclusion in the spanning tree. Considerations in the efficient implementation of such algorithms are examined and several heuristics for assigning node weights are compared.

On building minimum cost communication networks over time

Abstract: This paper deals with the problem of minimizing the present worth of expenditures needed to enlarge the capacity of a communications network over time so as to meet rising traffic demands. Eight general “tests” are given for improving network solutions. The tests require only a moderate amount of computation and have the property that they will improve a wide class of non-optimal solutions. It is observed that for a large subclass of practical problems, the dynamic problem of building a network over time may be converted to a static problem for most of the computations. This suggests a strong similarity between the optimal solutions to both problems.

Shortest path with time constraints on movement and parking

Abstract: The shortest path problem is well known and a number of methods have been proposed for its solution. In its simplest form no restrictions or special costs are placed on movement in the network. Some variations of the problem have been presented and solved, such as requirements for visiting specified nodes, turn penalties and prohibitions and time-dependent length of arcs. In this paper we present and solve the case where we seek the shortest path when some arcs are closed for travelling during specified periods of time. We allow parking in the vertices of the network, when it is necessary to wait for an arc to be opened, but we also assume the possibility of “no-parking” or “occupied” periods in the nodes. This type of problem emerges, for example, in the management of railway system, or a network of narrow roads with convoys moving on it.

Modeling and optimal control of congested transportation systems

Abstract: An approach is discussed for the modeling of congested transportation systems as store-and-forward networks analogous to communication networks of this type but with a finite travel time between nodes. A variety of approaches is suggested, and illustrated by examples, for the allocation of the usage of the arcs of the network by various competing units of traffic.

Generalized Ramsey theory for graphs IV, the Ramsey multiplicity of a graph

Abstract: A Proper graph G has no isolated points. Its Ramsey number r(G) is the minimum p such that every 2-coloring of the edges of Kp contains a monochromatic G. The Ramsey multiplicity R(G) is the minimum number of monochromatic G in any 2-coloring of Kr(G). With just one exception, namely K4, we determine R(G) for proper graphs with at most 4 points. For the stars K1,n, it is shown that R = 2n when n is odd and R = 1 when n is even. We conclude with the conjecture that for a proper graph, R(G) = 1 if and only if G = K2 or K1,n with n even.

Matchings in Polytopal Graphs.

Abstract: : A matching M of a graph G is a set of edges of G such that each vertex of G is incident with a most one edge of M, and that no edge of G may be added to M without violating this restriction. A number of new results on matchings are obtained, in particular concerning planar and polytopal graphs. A typical result is the following one which deals with (pi bar)(G), the largest possible number of edge-disjoint matchings of G. Theorem: The largest value of (pi bar)(G), possible for 3-polytopal graphs G is 12. (Author)

An algorithm for nonlinear multicommodity flow problems

Abstract: A nonlinear programming algorithm is applied to multicommodity flow problems with nonlinear costs. The algorithm can handle a wide class of problems and unifies several seemingly diverse approaches previously published. Significant computational simplifications that occur under special conditions are discussed.

Network properties of the ARPA computer network

Abstract: The ARPA Computer Network, which today links over 60 host computers in more than 40 different locations is a significant advance both in computer systems and data communications. The ARPANET uses a communication technique called packet switching to dynamically select message routes through the network. In this paper, we discuss a number of important properties of the network as well as the techniques used to identify these properties. Among the topics discussed are ARPANET growth, performance sensitivity to traffic variations, incremental costs for adding capacity and new sites, peak throughput costs, and network reliability.

Dynamic network flows with arc changes

Abstract: This paper modifies Ford and Fulkerson's Maximum Dynamic Flow Algorithm so that additional arcs can be added to or deleted from the network at a later time period.

Cost allocation in networks: The bulk supplier problem

Abstract: We consider the cost allocation problem faced by a large scale supplier (the telephone company) who rents telephone circuits to its customers. A number of possible allocation rate structures are analyzed, as are some of the criteria that they may be measured by. Finally, we draw conclusions and make recommendations for cost allocation in these circumstances.

Shortest alternating path algorithms

Abstract: The concept of an alternating path has been very useful in analyzing matching problems and has formed the basis for a number of matching algorithms. However, no techniques have been devised to find the shortest alternating path in a weighted graph. This paper defines different types of directed and undirected alternating paths, and shows how the problem of finding the shortest directed alternating path can be transformed into a problem of finding the shortest path in a directed graph. Utilizing this transformation, an efficient algorithm is developed for finding the shortest undirected alternating path. Computational experience is given. Extensions of the techniques in this paper to other types of alternating paths are discussed.

On clique‐extremal (p,q)‐graphs

Abstract: A clique of a graph is a maximal complete subgraph. A (p,q)-graph has p points and q lines. A clique-extremal (p,q)-graph has either the maximum or the minimum number of cliques among all (p,q)-graphs. Moon and Moser have determined constructively the maximum number of cliques in a p-point graph. The problem of studying the clique-extremal (p,q)-graphs is now investigated. We first develop a standard form for such extremal graphs. These standard forms are shown to have a complementary kind of analogy with respect to maximum and minimum.

Control of an expanding network—“An operational nightmare”

Abstract: Control of a network, especially as it expands into new geographic areas, is one of the most important factors in assuring that both the reliability and availability of that network will not be adversely effected. When the General Electric Company first offered a commercial network as part of its Time-Sharing service, only a small part of communications control was automated. The remainder of the control functions were performed manually. As the network grew from its infancy into what it is today, it could have become “An Operational Nightmare”. It became evident that it was necessary to develop into the network a completely automatic control system that would provide operations personnel with the tools for both recognizing and recovering from problems that might compromise either its availability or reliability. Included in this capability are the following: 1 The ability to detect and display the status of all the nodes. 2 The ability to detect and display the status of all links. 3 The ability to introduce new software. 4 The ability to measure the overall network performance. 5 The ability to schedule preventive maintenance. 6 The ability to detect imminent failures. This paper will show why this automated control system is necessary, how it is structured to record the pulse and heart-beat of the network and finally how it is used to maintain the stability of the network.

A multicommodity flow problem

Abstract: This paper presents an algorithm for finding maximal sized sets of flows in a certain class of multicommodity flow networks. The class consists of networks with integer capacity edges and with each node being a source or sink for all but at most one commodity. This new procedure lifts a restriction in a previous algorithm that required flow in edges joining source and sink of the same commodity.