Showing papers in "Networks in 1987"
TL;DR: The problem of determining a minimum cost connected network G that spans a given subset of vertices is known in the literature as the Steiner problem in networks and exact algorithms and heuristics are surveyed.
Abstract: The problem of determining a minimum cost connected network (i.e., weighted graph) G that spans a given subset of vertices is known in the literature as the Steiner problem in networks. We survey exact algorithms and heuristics which appeared in the published literature. We also discuss problems related to the Steiner problem in networks.
732 citations
TL;DR: Two bipartite matching problems arising in Vehicle Scheduling are considered and a heuristic algorithm based on Lagrangean relaxation for the capacitated version of the multicommodity matching is presented together with experimental results.
Abstract: Two bipartite matching problems arising in Vehicle Scheduling are considered: the capacitated matching and the multicommodity matching. For the former, given a reasonable cost structure, we can exhibit a polynomial time algorithm, while the general case is conjectured to be NP-hard. The latter problem is shown to be NP-hard. A heuristic algorithm based on Lagrangean relaxation for the capacitated version of the multicommodity matching is also presented together with experimental results.
222 citations
TL;DR: Sharp bounds on the cardinality of locating-dominating sets for arbitrary graphs on p vertices and for trees on p trees are given, and a linear algorithm for finding a minimum cardinality locating-Dominating set in an acyclic graph is presented.
Abstract: Locating-dominating sets are of interest in safeguard applications of graphical models of facilities. A subset S of the vertex set V of a graph G is a dominating set if each vertex u ϵ V - S is adjacent to at least one vertex in S. For each v in V - S let S(v) denote the set of vertices in S which are adjacent to v. A dominating set S is defined to be “locating” if for any two vertices v and w in V - S one has S(v) ≠ S(w). Sharp bounds on the cardinality of locating-dominating sets for arbitrary graphs on p vertices and for trees on p vertices are given, and a linear (that is O(P)) algorithm for finding a minimum cardinality locating-dominating set in an acyclic graph is presented.
211 citations
TL;DR: This article analyzes a special case of that problem, where the set of nodes, which must be included in the solution tree, consists of a single node, and all node weights are negative.
Abstract: The general Node-Weighted Steiner Tree problem is an extension of the standard Steiner Tree problem by the addition of node-associated weights. This article analyzes a special case of that problem, where the set of nodes, which must be included in the solution tree, consists of a single node, and all node weights are negative. The special case is shown to be NP-Complete, its integer programming formulation is presented, and heuristic procedures are proposed. Using Lagrangian relaxation and subgradient optimization, tight lower bounds were derived and utilized by a branch and bound algorithm. The effectiveness of the developed procedures is demonstrated by a set of computational experiments.
130 citations
TL;DR: In this article, the PartAN variant of the linear approximation method is adapted for solving the network equilibrium problem and its properties are analyzed by using algebraic and geometric approaches, and a simple and efficient algorithm is stated.
Abstract: The PARTAN variant of the linear approximation method is adapted for solving the network equilibrium problem. A simple and efficient algorithm is stated. Its properties are analyzed by using algebraic and geometric approaches. Its computational efficiency on small and large scale problems is compared to that of the linear approximation method.
84 citations
TL;DR: On developpe un algorithme en temps lineaire pour determiner si un graphe planaire triangule donne possede un dual rectangulaire.
Abstract: On developpe un algorithme en temps lineaire pour determiner si un graphe planaire triangule donne possede un dual rectangulaire
74 citations
TL;DR: The problem of finding the minimum length of feasible postman tour is NP-complete because the precedence relation on sets of arcs is general.
Abstract: Since the introduction of the Chinese Postman Problem (CPP), many variations on the same theme have been developed. In this paper we examine still another variation. The arcs of the graph are partitioned and a precedence relation defined, specifying the order in which the elements of the partition have to be traversed. We first examine the conditions for a feasible solution to the problem. Next, we specify the graph properties of the precedence partition that insure a polynomial complexity solution of O(N5), where N is the number of nodes in the original graph. When the precedence relation on sets of arcs is general, we prove that the problem of finding the minimum length of feasible postman tour is NP-complete.
66 citations
TL;DR: A tree generation algorithm is devised that exploits the special structure of the Steiner network problem while solving its constrained minimal spanning tree formulation and produces asymptotically optimal reduction.
Abstract: The Steiner network problem seeks a minimum weight connected subgraph that spans a specified subset off nodes in a given graph and optionally uses any of the other nodes as intermediate or Steiner points. This model has a variety of practical applications particularly in location, transportation, and communication planning. In this paper, we first outline some distinctive characteristics of optimal Steiner network solutions and propose a problem reduction procedure based on these properties. We derive a conservative estimate of the expected reduction achieved by this method under one set of probabilistic assumptions and demonstrate that this scheme produces asymptotically optimal reduction. We also report computational results for several randomly generated test problems. We then devise a tree generation algorithm that exploits the special structure of the Steiner network problem while solving its constrained minimal spanning tree formulation.
51 citations
TL;DR: A restricted minimum spanning tree model is adjusted for the NSP as well as for the Steiner Forest Problem, a newly introduced generalization.
Abstract: The Steiner Problem in Graphs (SP) is the problem of finding a set of edges with minimum total weight which connects a given subset of nodes in an edge-weighted (undirected) graph. In the more general Node-weighted Steiner Problem (NSP) also node weights are considered. A restricted minimum spanning tree model is adjusted for the NSP as well as for the Steiner Forest Problem, a newly introduced generalization. The NSP is related to the Directed Steiner Problem. Reduction tests for the SP, reducing the size of the problem graph, are adapted for these generalizations and some new tests are developed.
51 citations
TL;DR: The present paper contains polynomial time algorithms for finding maximum k-colorings andmaximum k-coverings of transitive graphs.
Abstract: Consider a graph G and a positive integer k. The maximum k-coloring problem is to color a maximum number of vertices using k colors, such that no two adjacent vertices have the same color. The maximum k-covering problem is to find k disjoint cliques covering a maximum number of vertices. The present paper contains polynomial time algorithms for finding maximum k-colorings and maximum k-coverings of transitive graphs.
TL;DR: A new and much faster algorithm is presented for the problem of assigning buses to a large number of short trips in an urban area, based on the hungarian method and making full use of the sparsity of the assignment matrix for the bus scheduling problem.
Abstract: A new and much faster algorithm is presented for the problem of assigning buses to a large number of short trips in an urban area. The trips are grouped into chains, beginning and ending at the same bus depot, and a vehicle is assigned to each one of them. Fleet size costs and dead heading time are to be minimized. This problem has been already formulated as a transportation problem and more recently, as an assignment model. However, some difficulties, such as the zero pivot phenomenon, rising in many practical cases drastically affected computing times required to obtain the optimal solution. This is overcome by an algorithm based on the hungarian method and making full use of the sparsity of the assignment matrix for the bus scheduling problem. Computational results comparing the different methods are given in the last section of the paper. Significant reductions in computing time are obtained for either real case applications or random generated test problems.
TL;DR: A polynomial-time algorithm is developed to obtain optimal solutions for the special case of parallel graphs and a heuristic algorithm is derived to solve some Very Large Scale Integrated Circuits (VLSI) design linear placement problems.
Abstract: A linear placement technique that uses an objective function of the sum of wiring lengths is proposed. The method evolves from well-known concepts in job sequencing and network flow. The relation between a job sequencing problem and this linear placement problem was demonstrated by Lawler. Also, Sidney proposed decomposition algorithms for job sequencing problems. Building on Lawler's and Sidney's work, we first develop a polynomial-time algorithm to obtain optimal solutions for the special case of parallel graphs. Adolphson and Hu applied the max-flow min-cut method of the network flow problem to the partitioning of general placement problems. However, when the cut operation creates the cut of the same configuration as the previous cut operations, no additional partitioning information is obtained. We devise an optimal graph modification that tries to change the configuration of the cut for further partitioning of the problem, and achieve the maximum partitioning of the problem. Finally, a heuristic algorithm is derived to solve some Very Large Scale Integrated Circuits (VLSI) design linear placement problems. A comparison with published papers shows that our VLSI placement method produces better results.
TL;DR: A sampling plan for estimating {Mk} and linear functions of these parameters, including the T-connectedness reliability function g(p) for common failure probability 1 - p, is described and a priori bounds are derived.
Abstract: Consider an undirected network G with node set V and arc set E = {1, …, n} where arcs fail randomly and independently. Let T be a subset of V and let Mk denote the number of ways that all the nodes of T are connected (T-connectivity) with exactly k operating arcs and n - k failed arcs. This paper describes a sampling plan for estimating {Mk} and linear functions of these parameters, including the T-connectedness reliability function g(p) for common failure probability 1 - p. Point and simultaneous interval estimates are derived for {Mk/(}, where the interval estimates meet a fixed width criterion in O(n log n) time as the size of the network n grows. Whereas all previously proposed Monte Carlo sampling plans enable one to estimate g(p) for a fixed p, the proposed method allows one to estimate the entire function {g(p), p ∈ P}, where either P = P* = {qi: O ≦ qi ≦ 1, 1 ≦ i = ≦ v} or P = P** = {[a, b]: 0 ≦ a ≦ b ≦ 1}. Here P* denotes a finite set of points in [0, 1] and P** an interval in [0, 1]. Simultaneous interval estimates are derived that meet a fixed width criterion in O(n) time for P = P* and in O(n2) time for P = P** as n ∞. A priori bounds are also derived for g (p) and it is shown how these can be used to give guidance on the performance of the sampling plan. An example based on a network with 44 nodes and 85 arcs illustrates the proposed method.
TL;DR: For a red bipartite graph with given cardinality, the problem of compacting a programmable logic array is formulated as the following graph problem, and a polynomial heuristic algorithm gives an optimum solution.
Abstract: The problem of compacting a programmable logic array is formulated as the following graph problem. Given a red-edge bipartite graph, how to add maximum number of independent green edges such that there are no cycles formed by alternating red and green edges. For this NP-complete problem, we present a polynomial heuristic algorithm which gives an optimum solution when the red bipartite graph satisfies certain conditions, e.g., a tree. When the bipartite graph does not satisfy these conditions, the heuristic algorithm gives a solution with worst-case error bound. For a red bipartite graph with given cardinality, we give a tight upper bound on the number of green edges.
TL;DR: Most of the results are produced for Euclidean TSP's, but evidence is presented that indicates that the results apply equally well if not more strongly to the general symmetric TSP.
Abstract: A method for accelerating the computational performance of branch exchange heuristics for symmetric traveling salesman problems (TSP's) is presented. The improvement in performance is obtained by considering only exchanges that have a good chance of producing a better solution. In the instance of the 3–optimal heuristic, the approach reduces the number of operations required to obtain a good solution to a TSP with N nodes from O(N3) to O(N2), without a corresponding degradation in the quality of the solution. Most of the results are produced for Euclidean TSP's, but evidence is presented that indicates that thes results apply equally well if not more strongly to the general symmetric TSP.
TL;DR: In this paper, two versions of the generalized Dijkstrauss algorithm are compared and the advantage of the adaptive version grows with both k and the network size, and the adaptive algorithm exhibits a 100:1 advantage in the number of permanent labels set.
Abstract: In this paper, we examine the problem of finding k-shortest paths between an origin and destination pair when distances are Euclidean. Two versions of a generalized Dijkstra algorithm are compared. Computational results show that the advantage of the adaptive version (measured by total number of permanent labels) grows with both k and the network size. For large networks, the adaptive algorithm exhibits a 100:1 advantage in the number of permanent labels set.
TL;DR: This work shows how the problem of investing in new arcs in such a network in order to increase the expected max flow as much as possible can be formulated as a stochastic program with network recourse.
Abstract: Consider a network with arcs subject to failures. We show how the problem of investing in new arcs in such a network in order to increase the expected max flow as much as possible can be formulated as a stochastic program with network recourse. We show how to decompose the problem, and consider both exact methods and approximations. Convergence proofs are given. We demonstrate that max flow recourse problems can be solved very efficiently, since lower and upper bounds are equally simple to evaluate.
TL;DR: This paper presents certain generalizations and improvements of Hajek and Sasaki's results on the scheduling of data transfers in networks where interruption does not permit interruption and each communication module can be used as a transmitter and as a receiver.
Abstract: The scheduling of data transfers in networks, where the schedule does not permit interruption and each communication module can be used as a transmitter and as a receiver (i.e., as a transceiver) was studied by Coffman et al. The same problem when interruption in the schedule is permitted and the transmitting and receiving modules are distinct was studied by Choi and Hakimi among others. Hajek and Sasaki studied another interesting variation of the problem where interruption is permitted but each communication module is a transmitter and a receiver. This paper presents certain generalizations and improvements of Hajek and Sasaki's results.
TL;DR: This paper proves that the determination of la(G), the minimum number of linear diforests which partition the arcs of G, is NP-complete and also that the problem becomes polynomial for acyclic digraphs.
Abstract: A linear diforest is a digraph whose connected components are paths. We define the linear arboricity of a digraph, denoted la(G), as the minimum number of linear diforests which partition the arcs of G. In this paper, we prove that the determination of la(G) is NP-complete. We prove also that the problem becomes polynomial for acyclic digraphs. Finally, we give the value of la(G) for several families of digraphs, in particular 2-regular digraphs.
TL;DR: On montre que trouver une coupe minimum ou un flot maximum dans un reseau acyclique n'est pas plus facile que le meme probleme dansun reseau general.
Abstract: On montre que trouver une coupe minimum ou un flot maximum dans un reseau acyclique n'est pas plus facile que le meme probleme dans un reseau general
TL;DR: This paper investigates the investigation of minimal graphs that are k-regular in addition to being k neighbor-connected, and gives some results connecting the minimal graphs to other combinatorial objects.
Abstract: In [G. Gunther, Neighbor-connectivity in regular graphs. Discrete Appl. Math.11 (1985) 233–243] Gunther introduced the concept of a k neighbor-connected graph, which has the property that the removal of any k − 1 closed neighborhoods neither disconnects the graph, nor leaves only a complete graph. In this paper we pursue the investigation of minimal graphs that are k-regular in addition to being k neighbor-connected. In a private communication, Gunther conjectured that if G is such a graph which contains no cliques of size larger than m, then |V(G)| ≧ k2 + (k + 1 − m) (k − 1) + 1. In the above reference, he proved that this conjecture is valid in the case that m = k and characterized the minimal graphs. In this paper, we begin to investigate the case where m = 2. We give some results connecting the minimal graphs to other combinatorial objects.
TL;DR: Series-parallel graphs, outerplanar graphs, and graphs whose polygon matroids are transversal have been characterized by forbidden subgraphs are related to properties of the graph decomposition.
Abstract: Series-parallel graphs, outerplanar graphs, and graphs whose polygon matroids are transversal have been characterized by forbidden subgraphs. Tutte introduced a graph decomposition for nonseparable graphs. The results of this paper relate the existence of the forbidden subgraphs to properties of the decomposition. Algorithmic implications are considered.
TL;DR: Two new lower planes for the network design problems through combinatorial arguments are derived and one is of complexity O(n4) and produces a lower bound which is sharper than those of existing lower planes.
Abstract: A lower plane for the network design problem is a linear lower approximation of the objective function and is used to obtain a lower bound in the branch and bound algorithm. In this article, we derive two new lower planes for the network design problems through combinatorial arguments. The first lower plane is of complexity O(n4) and produces a lower bound which is sharper than those of existing lower planes. The second lower plane is of complexity O(n3) and produces a reasonably good lower bound. Computational results are presented comparing the bounds obtained by the new lower planes with those of the existing lower planes.
TL;DR: A digraph is constructed with the maximum number of simple paths between two specified vertices, for a digraph with a given number of edges, which are the tri-chains, the (deficient) Fibonacci digraphs and (if the conjecture is true) the 3-diamond strings.
Abstract: We construct a digraph with the maximum number of simple paths between two specified vertices, for a digraph with a given number of edges. The following cases are considered: digraphs with parallel edges, acyclic simple digraphs and general simple digraphs. The corresponding extremal digraphs are the tri-chains, the (deficient) Fibonacci digraphs and (if our conjecture is true) the 3-diamond strings, respectively. The similarity of these three families of digraphs is discussed. The related problem of digraphs with the maximum number of simple cycles for a given number of edges is considered too.
TL;DR: A technique producing minimum buffer graphs for a rather large class of networks is illustrated, and a comparison is made with other well-known deadlock avoidance techniques, showing that substantial savings can be achieved.
Abstract: Store-and-forward communication networks may be designed so as to be free from store-and-forward deadlock. This is accomplished by incorporating in the network an acyclic buffer graph on which messages are forwarded, from buffer to buffer, according to all the desired routes. A technique producing minimum buffer graphs for a rather large class of networks is illustrated. Successively, a comparison is made with other well-known deadlock avoidance techniques, showing that substantial savings can be achieved.
TL;DR: The purpose of this paper is to introduce concepts which make it possible to generalize this algorithm to some classes of hypergraphs, by providing a polynomial primal-dual algorithm for the matching problem inhypergraphs without odd cycles.
Abstract: The matching problem in bipartite graphs can be solved by an elegant primal-dual algorithm. The purpose of this paper is to introduce concepts which make it possible to generalize this algorithm to some classes of hypergraphs. We illustrate the approach by providing a polynomial primal-dual algorithm for the matching problem in hypergraphs without odd cycles.
TL;DR: The theorem which gives the essential relationship between the SC and Di- SC networks is proved and the Odd-Addition-Subtraction (OAS) condition is given as a necessary condition for Di-SC functions.
Abstract: The theory of directed switching networks is developed. After the fundamental concepts are defined, the rank of the path matrix of a Directed Single-Contact (Di-SC) network is discussed. The theorem which gives the essential relationship between the SC and Di-SC networks is proved. Then the Odd-Addition-Subtraction (OAS) condition is given as a necessary condition for Di-SC functions. The analysis of directed switching networks and the synthesis of Di-SC networks are also given for practical applications. The results could be used in the design of electrical and computer systems with directed elements.