Showing papers in "Networks in 1981"
TL;DR: This paper presents a heuristic for this problem in which an assignment of customers to vehicles is obtained by solving a generalized assignment problem with an objective function that approximates delivery cost and shows that it has outperformed the best existing heuristics on a sample of standard test problems.
Abstract: : We consider a common variant of the vehicle routing problem in which a vehicle fleet delivers products stored at a central depot to satisfy customer orders. Each vehicle has a fixed capacity, and each order uses a fixed portion of vehicle capacity. The routing decision involves determining which of the demands will be satisfied by each vehicle and what route each vehicle will follow in servicing its assigned demand in order to minimize total delivery cost. We present a heuristic for this problem in which an assignment of customers to vehicles is obtained by solving a generalized assignment problem with an objective function that approximates delivery cost. This heuristic has many attractive features. It has outperformed the best existing heuristics on a sample of standard test problems. It will always find a feasible solution if one exists, something no other existing heuristic can guarantee. It can be easily adapted to accommodate many additional problem complexities. By parametrically varying the number of vehicles in the fleet, our method can be used to optimally solve the problem of finding the minimum size fleet that can feasibly service the specified demand.
1,050 citations
TL;DR: In this paper, the complexity of a class of vehicle routing and scheduling problems is investigated, and the results on the worst-case performance of approximation algorithms are discussed and some directions for future research are suggested.
Abstract: The complexity of a class of vehicle routing and scheduling problems is investigated. We review known NP-hardness results and compile the results on the worst-case performance of approximation algorithms. Some directions for future research are suggested. The presentation is based on two discussion sessions during the Workshop to Investigate Future Directions in Routing and Scheduling of Vehicles and Crews, held at the University of Maryland at College Park, June 4–6, 1979.
1,017 citations
TL;DR: The intent in this paper is to define a capacitated arc routing problem, to provide mathematical programming formulations, to perform a computational complexity analysis, and to present an approximate solution strategy for this class of problems.
Abstract: A capacitated node routing problem, known as the vehicle routing or dispatch problem, has been the focus of much research attention On the other hand, capacitated arc routing problems have been comparatively neglected Both classes of problems are extremely rich in theory and applications Our intent in this paper is to define a capacitated arc routing problem, to provide mathematical programming formulations, to perform a computational complexity analysis, and to present an approximate solution strategy for this class of problems In addition, we identify several related routing problems and develop tight lower bounds on the optimal solution
519 citations
TL;DR: This paper gives a survey of a general relaxation procedure whereby the state-space associated with a given dynamic programming recursion is relaxed in such a way that the solution to the relaxed recursion provides a bound which could be embedded in general branch and bound schemes for the solution of the problem.
Abstract: It is well-known that few combinatorial optimization problems can be solved effectively by dynamic programming alone, since the number of vertices of the state space graph is enormous. What we are proposing here is a general relaxation procedure whereby the state-space associated with a given dynamic programming recursion is relaxed in such a way that the solution to the relaxed recursion provides a bound which could be embedded in general branch and bound schemes for the solution of the problem. This state space relaxation method is analogous to Langrangian relaxation in integer programming. This paper gives a survey of this new methodology, and gives, as examples, applications to the traveling salesman problem (TSP), the timeconstrained TSP and the vehicle routing problem (VRP). Valid state space relaxations are discussed for these problems and several bounds are derived in each case. Subgradient optimization and “state space ascent” are discussed as methods of maximizing the resulting lower bounds. More details of the procedures surveyed in this paper can be found in [2 ,3 ,41.
338 citations
330 citations
TL;DR: This discussion shows that prospects for applying exact methods, possibly in conjunction with heuristics, are far from fully realized and points to vehicle fleet planning as a tempting target of opportunity for further investigation.
Abstract: As a well-structured and costly activity that pervades industries in both the public and private sector, vehicle fleet management would appear to be a splendid candidate for model-based planning and optimization. And yet, until recently the combinatorial intricacies of vehicle routing and of vehicle scheduling have precluded the widespread use of optimization (exact) methods for this problem class. Our discussion in this paper identifies the extent and nature of these problem complexities and draws contrasts with other applications of combinatorial optimization. It also summarizes a number of successful uses of optimization for vehicle fleet planning and highlights potentially fruitful avenues for algorithmic development. In particular, we describe several alternative models and novel algorithms for the vehicle routing problem, show how various modeling approaches for this problem are intimately related, and illustrate the interplay between model formulations and the algorithms that they suggest. This discussion shows that prospects for applying exact methods, possibly in conjunction with heuristics, are far from fully realized and points to vehicle fleet planning as a tempting target of opportunity for further investigation.
181 citations
TL;DR: This work classifies the features that seem to be encountered in real vehicle routing problems and indicates which features cause the greatest difficulty and which modeling approaches allow to represent the greatest range of practical considerations or features.
Abstract: We classify the features that seem to be encountered in real vehicle routing problems. Given these features, we try to indicate which features cause the greatest difficulty and which modeling approaches allow us to represent the greatest range of practical considerations or features.
169 citations
TL;DR: The human aided optimization procedure was tested on the standard 50- point, 75-point, and 100-point test problems of Eilon, Watson-Gandy, and Christofides and a better solution was generated than the current best known solution.
Abstract: The set partitioning model is used as the basis for an interactive approach for solving a broad class of routing problems. A pricing mechanism is developed which can be used with a variety of methods in generating improving solutions. A version of the approach for delivery problems has been implemented via a colorgraphics display. The human aided optimization procedure was tested on the standard 50-point, 75-point, and 100-point test problems of Eilon, Watson-Gandy, and Christofides [6]. In the case of the first two test problems, the procedure was able to generate the best known solutions. In the 100-point problem, a better solution was generated than the current best known solution.
153 citations
TL;DR: This paper focuses on recent developments that have made this model more attractive and have resulted in several successful implementations, as well as new solution techniques employing Lagrangian relaxation and subgradient optimization.
Abstract: The set partitioning model of the crew rotation problem has been well known for many years. This paper focuses on recent developments that have made this model more attractive and have resulted in several successful implementations. These developments include improved problem conceptualizations and decompositions, as well as new solution techniques employing Lagrangian relaxation and subgradient optimization. Experience is reported from The Flying Tiger Line, Pacific Southwest Airways, Continental Airlines, and Helsinki City Transport. A case is made for work on heuristic decomposition methods to break large problems into moderate sized pieces that can be solved exactly.
137 citations
TL;DR: An efficient algorithm for finding minimal distance feasible paths between the points, assuming that all travel occurs according to the rectilinear distance metric is developed.
Abstract: Given a set of origin-destination points in the plane and a set of polygonal barriers to travel, this paper develops an efficient algorithm for finding minimal distance feasible paths between the points, assuming that all travel occurs according to the rectilinear distance metric. By geometrical arguments the problem is reduced to a finite network problem. The nodes are the origin-destination points and the barrier vertices. The links designate those node pairs that ”communicate” in a simple way, where communication implies the existence of a node-to-node rectilinear path that is not made longer by the barriers. The weight of each link is the rectilinear distance between its two corresponding nodes. Solution of the minimal distance path problem on the network procedes in two steps. First, for a given origin or root node, a tree is generated containing a minimal distance path to each node that communicates with the root node. Second, a modified Diikstra-type iteration is utilized, starting with the nodes of the tree, sequentially adding nodes according to minimum “penalty distance,” where the penalty is the extra travel distance caused by the barriers. The paper concludes with a discussion of the computational complexity of the procedure, followed by a numerical example.
132 citations
TL;DR: A polynomial time algorithm for testing isomorphism of permutation graphs (comparability graphs of 2-dimensional partial orders) is described and performs two types of simplifying transformations on the graph.
Abstract: A polynomial time algorithm for testing isomorphism of permutation graphs (comparability graphs of 2-dimensional partial orders) is described. It operates by performing two types of simplifying transformations on the graph; the contraction of duplicate vertices and the contraction of uniquely orientable induced subgraphs.
TL;DR: The linear arboricity of a graph is the minimum number of linear forests into which its lines can be decomposed and it is found that every 4-regular graph is 3.
Abstract: The linear arboricity of a graph is the minimum number of linear forests into which its lines can be decomposed. We find that the linear arboricity of every 4-regular graph is 3. This result enables us to obtain bounds for the linear arboricity of any graph in terms of its maximum degree.
TL;DR: An O(n log n) heuristic for the Euclidean Steiner Minimal Tree (ESMT) problem is presented and is shown to be as good as the previous O( n4) algorithm in achieving reductions in the ratio SMT/MST of the given vertex set.
Abstract: An O(n log n) heuristic for the Euclidean Steiner Minimal Tree (ESMT) problem is presented. The algorithm is based on a decomposition approach which first partitions the vertex set into triangles via the Delaunay triangulation, then “recomposes” the suboptimal Steiner Minimal Tree (SMT) according to the Voronoi diagram and Minimum Spanning Tree (MST) of the point set. The ESMT algorithm was implemented in FORTRAN-IV and tested on a number of randomly generated point sets in the plane drawn from a uniform distribution. Comparison of the O(n log n) algorithm with an O(n4) algorithm clearly indicates that the O(n log n) algorithm is as good as the previous O(n4) algorithm in achieving reductions in the ratio SMT/MST of the given vertex set. This is somewhat surprising since the O(n4) algorithm considers more potential Steiner points and alternative tree configurations.
TL;DR: The results indicate that the shortest augmenting path method is superior with respect to running time and that even larger problems may be solved in a reasonable amount of time.
Abstract: An efficient procedure for solving minimum weight perfect matching problems is presented. Starting from the empty matching the optimal matching is constructed by successively augmenting along shortest augmenting paths. Such paths can be determined via a special labeling technique. The algorithm is motivated by purely combinatorially natured optimality criteria using the concept of admissible transformations of the cost coefficients. We report on some experience with computer implementations of two different versions of this method and an implementation of Edmonds' BLOSSOM-algorithm which makes use of Lawler's labeling technique. Though all three methods are comparable with respect to computational complexity the results indicate that the shortest augmenting path method is superior with respect to running time and that even larger problems may be solved in a reasonable amount of time.
TL;DR: It is shown that the persistence of a graph of diameter d is the minimum over all pairs of nonadjacent points of the maximum number of disjoint paths of length at most d joining them.
Abstract: It is well known that the maximum connectivity k of a graph G with p points and q lines is given by [2 q/p]. This is restated in two useful alternative forms which minimize q given p and k, and which maximize p in terms of q and k. We define the persistence of a graph as the smallest number of points whose removal increases the diameter. It is shown that the persistence of a graph of diameter d is the minimum over all pairs of nonadjacent points of the maximum number of disjoint paths of length at most d joining them. A similar result is obtained for line-persistence and it is shown that these invariants are independent of each other.
TL;DR: NP-hardness for the design problem of finding a subgraph F of G which connects up all the original vertices and minimizes the maximum distance in F, subject to a budget constraint on the sum of the edge costs is established.
Abstract: Given a graph G with edge lengths and costs, we wish to find a subgraph F of G which connects up all the original vertices and minimizes the maximum distance in F, subject to a budget constraint on the sum of the edge costs. In this note we establish NP-hardness for the design problem, even for the simple case where G is a planar graph with maximum degree 3 and the budget restricts the choice to spanning trees. Moreover, the problem of finding a near optimal subgraph F is also NP-hard.
TL;DR: It is shown that 2-trees are minimum IFI networks and an algorithm is described which adds lines to an arbitrary tree network to produce an IFI network (a 2-tree) which determines routing tables which allow a simple calling protocol to complete all message transfers between operative sites under isolated failures.
Abstract: The notion of isolated failure immune (IFI) networks is introduced. A network is an IFI network if and only if all message transfers between operating sites can be completed as long as all site and line failures are isolated. Failures occurring at any time are isolated if they are pairwise isolated. A pair of site failures is isolated if the down sites are not neighbors. A pair of line failures is isolated if the lines are not incident to a common site. A site and a line failure are isolated if the down line is not incident to a neighbor of the down site. We show that 2-trees are minimum IFI networks. An algorithm is described which adds lines to an arbitrary tree network to produce an IFI network (a 2-tree). The algorithm also determines routing tables which allow a simple calling protocol to complete all message transfers between operative sites under isolated failures. Specializations and generalizations of our notion of IFI networks are discussed and issues for future research are proposed.
TL;DR: The adaptation of the dual simplex algorithm to compute the shortest paths from node s results in an algorithm which has the flavor of a label-setting method, which generally does not require the examination of all the nodes of the network.
Abstract: We present an adaptation of the dual simplex algorithm, for computing all shortest paths on a network. Given a shortest path arborescence rooted at node r, the change of root to a new origin s, renders the arborescence rooted at r dual feasible and primal infeasible for the new problem. The adaptation of the dual simplex algorithm to compute the shortest paths from node s results in an algorithm which has the flavor of a label-setting method. It generally does not require the examination of all the nodes of the network. We report some computational results with the method which indicate that it is at least as efficient as successive applications of a label-setting or a label-correcting shortest path algorithm.
TL;DR: An algorithm which requires O(n2) steps to generate one maximum clique is presented and the algorithm can also be used to generate all maximum cliques where the number of steps need to generate each additional maximumClique is linear in its size.
Abstract: A circle diagram consists of a circle C and a set of n chords. This diagram defines a graph with n vertices where each vertex corresponds to a chord, and two vertices are adjacent if their corresponding chords intersect in C. A graph G is called a circle graph if it is defined by some circle diagram. An algorithm which requires O(n2) steps to generate one maximum clique is presented. The algorithm can also be used to generate all maximum cliques where the number of steps need to generate each additional maximum clique is linear in its size. This compares favourably with Gavril's algorithm [4] which works in O(n3) steps.
TL;DR: This discussion session was concerned with the analysis of heuristics and the design of more effective heuristic approaches and a classification of heuristic approaches was devised.
Abstract: This discussion session was concerned with the analysis of heuristics and the design of more effective heuristics. Initially, the group generated a list of evaluation criteria and discussed the strengths and weaknesses of each criterion. Next a classification of heuristic approaches was devised. Finally, the group compiled a list of promising research directions.
TL;DR: This paper presents a polynomial time algorithm for finding an absolute center of a network that is combinatorial in nature and requires only knowledge of the shortest path distances between all pairs of vertices.
Abstract: The absolute center of a network is any vertex or point on an edge such that the distance from it to the vertex farthest from it is as small as possible. This paper presents a polynomial time algorithm for finding an absolute center. This algorithm is combinatorial in nature and requires only knowledge of the shortest path distances between all pairs of vertices.
TL;DR: Two shortest path algorithms are compared and it is shown that, while one outperforms the other in practice, the former's running time is exponential in the worst case while the latter's is polynomial.
Abstract: Two shortest path algorithms are compared and it is shown that, while one outperforms the other in practice, the former's running time is exponential in the worst case while the latter's is polynomial. A procedure which constructs such worst case examples is given.
TL;DR: After stating the weighted, perfect matching problem and briefly describing Edmonds' algorithm, certain postoptimality procedures are described, which aid in reoptimizing related matching problems in which a few edge weights are altered.
Abstract: After stating the weighted, perfect matching problem and briefly describing Edmonds' algorithm, certain postoptimality procedures are described. These procedures aid in reoptimizing related matching problems in which a few edge weights are altered. Regardless of the actual implementation of the matching algorithm used, when changing a single edge weight, the postoptimality procedures are on the order of cardinality (N) more efficient than solving the modified problem “from scratch,” where N is the node set of the underlying graph.
TL;DR: The problem of updating shortest paths from all the vertices to a set of vertices when the length function is decreased can be solved by means of the Dijkstra method in a running time of the same order as Goto and Sangiovanni-Vincentelli's algorithm, assuming a smaller amount of initial data than theirs, i.e., the authors do not need the LU-factorization of the measure matrix.
Abstract: The problem of updating shortest paths from all the vertices to a set of vertices when the length function is decreased was considered by S. Goto and A. Sangiovanni-Vincentelli and a solution algorithm was presented based on the LU-factorization of the measure matrix and a matrix inversion formula. The present paper shows that the problem can be solved by means of the Dijkstra method in a running time of the same order as Goto and Sangiovanni-Vincentelli's algorithm, assuming a smaller amount of initial data than theirs, i.e., we do not need the LU-factorization of the measure matrix.
TL;DR: In this article, the authors apply the inclusion-exclusion principle of probability theory to the event that at least one rooted spanning tree of the graph is working, and prove combinatorial properties of graphs which allow them to derive a much condensed form of the inclusion exclusion expression.
Abstract: This paper is concerned with the problem of computing the probability that a root vertex can communicate with all other vertices in a probabilistic directed graph. One method is to apply the inclusion-exclusion principle of probability theory to the event “at least one rooted spanning tree of the graph is working.” We prove combinatorial properties of graphs which allow us to derive a much condensed form of the inclusion-exclusion expression. Each term corresponds to an acyclic spanning subgraph of the original graph, with coefficient equal to (−1)b-n+1, where b and n are the number of edges and vertices of the subgraph, respectively.
TL;DR: Although each of the algorithms has worst case complexity O(kp), where k and p are the number of cotree arcs and nodes, respectively, a variation of a root traceback algorithm is shown to be the fastest in almost all cases.
Abstract: Given a connected directed graph and a spanning tree, we consider the problem of finding the set of fundamental cycles. In particular, for each cotree arc i and tree arc j, we need to know whether or not i and j are in the same fundamental cycle, and if so, whether or not arcs i and j are oriented in the same direction. This problem has application in primal network flow, longest cycle, and all-cycle algorithms. In this paper, we describe and compare three algorithms for finding fundamental cycles. Computational results are presented on a variety of directed graphs produced by a network generator. Although each of the algorithms has worst case complexity O(kp), where k and p are the number of cotree arcs and nodes, respectively, a variation of a root traceback algorithm is shown to be the fastest in almost all cases.
TL;DR: In an earlier paper the measures of centrality in a graph G demonstrated by the center and centroid were unified and generalized, resulting in the definition of the h-centrum of G and denoted C(G;h) for 1 ≤ h ≤ |V(G)|.
Abstract: In an earlier paper the measures of centrality in a graph G demonstrated by the center and centroid were unified and generalized, resulting in the definition of the h-centrum of G and denoted C(G;h) for 1 ≤ h ≤ |V(G)|. When dealing with the h-centrum one sums the distances from a given vertex u to each of the h vertices farthest from it. Here another way of generalizing the concepts of center and centroid is examined, and the k-nucleus is defined for G and denoted θk(G). When dealing with the k-nucleus one sums the distances from a given vertex u to the balls of radius k around each of the other vertices. The centroid of G is θo(G), and the center of G is θr(G), where r is the radius of G. Some relationships between the h-centra and the k-nuclei are presented.
TL;DR: This paper provides a characterization of graphs quasi-prime with respect to cartesian product, as well as graphs quasiprime with respectto other products.
Abstract: A graph is quasiprime with respect to a boolean product of graphs if whenever it is a subgraph of the product of two graphs, it must necessarily be isomorphic to a subgraph of one of its factors. This paper provides a characterization of graphs quasi-prime with respect to cartesian product, as well as graphs quasiprime with respect to other products.
TL;DR: Formulas for series and parallel reductions are obtained for difficult measures of network reliability for systems analyzed by Monte-Carlo sampling, and it is found that the estimator variance is reduced by the replacement.
Abstract: Formulas for series and parallel reductions are obtained for difficult measures of network reliability. Examples considered include “traffic to center,” “total traffic carried,” “minimum cut,” and “all demanding vertices communicate.” We find general properties of “distribution-preserving” and “average-preserving” replacements, and present several examples. For systems analyzed by Monte-Carlo sampling, we find that the estimator variance is reduced by the replacement.
TL;DR: It is shown that if a point v of G has negative cohesiveness, then the set of points adjacent to v is the unique minimum size disconnecting set of G, and the necessary and sufficient conditions for an arbitrary triple to be the coheseness triple of a graph are derived.
Abstract: The connectivity contribution or cohesiveness of a point v of graph G is defined as the difference k(G) - k(G - v) where kappa is the usual connectivity symbol It is shown that if a point v of G has negative cohesiveness, then the set of points adjacent to v is the unique minimum size disconnecting set of G This theorem has several corollaries including the result that if v has negative cohesiveness in G, then it does not in Γ Finally we define a cohesiveness triple (n_, n0, n+) of a graph by taking these, respectively, as the number of negative, zero, and positive cohesiveness points of G The necessary and sufficient conditions for an arbitrary triple to be the cohesiveness triple of a graph are derived