Showing papers in "Networks in 1993"

Parallel iterative search methods for vehicle routing problems

Abstract: This paper presents two partition methods that speed up iterative search methods applied to vehicle routing problems including a large number of vehicles. Indeed, using a simple implementation of taboo search as an iterative search method, every best-known solution to classical problems was found. The first partition method (based on a partition into polar regions) is appropriate for Euclidean problems whose cities are regularly distributed around a central depot. The second partition method is suitable for any problem and is based on the arborescence built from the shortest paths from any city to the depot. Finally, solutions that are believed to be optimum are given for problems generated on a grid. © 1993 by John Wiley & Sons, Inc.

A Catalog of Steiner Tree Formulations

Abstract: We present some existing and some new formulations for the Steiner tree and Steiner arborescence problems. We show the equivalence of many of these formulations. In particular, we establish the equivalence between the classical bidirected dicut relaxation and two vertex weighted undirected relaxations. The motivation behind this study is a characterization of the feasible region of the dicut relaxation in the natural space corresponding to the Steiner tree problem. 0 7993 by John Wiley & Sons, Inc.

A new class of interconnection networks based on the alternating group

Abstract: This paper introduces a new class of interconnection scheme based on the Cayley graph of the alternating group. It is shown that this class of graphs are edge symmetric and 2-transitive. We then describe an algorithm for (a) packet routing based on the shortest path analysis, (b) finding a Hamiltonian cycle, (c) ranking and unranking along the chosen Hamiltonian cycle, (d) unit expansion and dilation three embedding of a class of two-dimensional grids, (e) unit dilation embedding of a variety of cycles, and (f) algorithm for broadcasting messages. The paper concludes with a short analysis of contention resulting from a typical communication scheme. Although this class of graphs does not possess many of the symmetry properties of the binary hypercube, with respect to the one source broadcasting, these graphs perform better than does a hypercube, and with respect to the contention problem, these graphs perform better than do the star graphs and are close to the hypercube. © 1993 by John Wiley & Sons, Inc.

A simple and fast label correcting algorithm for shortest paths

Abstract: We propose a new method for ordering the candidate nodes in label correcting methods for shortest path problems. The method is equally simple but much faster than the D’ Esopo-Pape algorithm. It is similar to the threshold algorithm in that it tries to scan nodes with small labels as early as possible, and performs comparably with that algorithm. Our algorithm can also be combined with the threshold algorithm thereby considerably improving the practical performance of both algorithms.

The traveling salesman problem with cumulative costs

Abstract: In this paper, we consider a special case of the time-dependent traveling salesman problem where the objective is to minimize the sum of all distances traveled from the origin to all other cities. Two exact algorithms, incorporating lower bounds provided by a Lagrangean relaxation of the problem, are presented. We also investigate a heuristic procedure derived from dynamic programming that is able to evaluate the distance from optimality of the produced solution. Computational results for a number of problems ranging from 15 to 60 cities are given. They show that problems up to 35 cities can be solved exactly and problems up to 60 cities can be solved within 3% from optimality. © 1993 by John Wiley & Sons, Inc.

Shortest paths, single origin‐destination network design, and associated polyhedra

Abstract: We study a specialized version of network design problems that arise in telecommunications, transportation, and other industries. The problem, a generalization of the shortest path problem, is defined on an undirected network consisting of a set of arcs on which we can install (load), at a cost, a choice of up to three types of capacitated facilities. Our objective is to determine the configuration of facilities to load on each arc that will satisfy the demand of a single commodity at the lowest possible cost. Our results (i) demonstrate that the single-facility loading problem and certain “common break-even point” versions of the two-facility and three-facility loading problems are polynomially solvable as a shortest path problem; (ii) show that versions of the two-facility loading problem are strongly NP-hard, but that a shortest path solution provides an asymptotically “good” heuristic; and (iii) characterize the optimal solution (i.e., specify a linear programming formulation with integer solutions) of the common break-even point versions of the two-facility and three-facility loading problems. In this development, we introduce two new families of facets, give geometric interpretations of our results, and demonstrate the usefulness of partitioning the space of the problem parameters to establish polyhedral integrality properties. Generalizations of our results apply to (i) multicommodity applications and (ii) situations with more than three facilities. © 1993 by John Wiley & Sons, Inc.

Additive graph spanners

Abstract: A spanning subgraph S = (V, E′) of a connected simple graph G = (V, E) is a f(x)-spanner if for any pair of nodes u and v, dS(u, v) ≦ f(dG(u, v)), where dG and dS are the usual distance functions in graphs G and S, respectively. We are primarily interested in (t + x)-spanners, which we refer to as additive spanners. We construct low-degree additive spanners for X-trees, pyramids, and multidimensional grids. We prove, for arbitrary t > 0, that to determine whether a given graph G has an additive spanner with no more than m edges is NP-complete. © 1993 by John Wiley & Sons, Inc.

A two-commodity flow formulation for the traveling salesman and the makespan problems with time windows

Abstract: We present a new two-commodity flow formulation for the traveling salesman problem. Each commodity corresponds to a resource that is either distributed or picked up along the tour of all nodes. This formulation is partcularly well suited to handling time window constraints; the resource used is then the time. This formulation can be extended to the makespan problem. For a n-node problem, the linear relaxation of the formulation involves only O(n) constraints and O(n2) variables. Implementation issues are discussed and numerical experimentations have been realized for problems of up to 60 nodes. © 1993 by John Wiley & Sons, Inc.

Fundamental algorithms for the star and pancake interconnection networks with applications to computational geometry

Abstract: The star and pancake networks were recently proposed as attractive alternatives to the hypercube topology for interconnecting processors in a parallel computer. However, few parallel algorithms are known for these networks. In this paper, we present several data communication schemes and basic algorithms for these two networks. These algorithms are then used to develop parallel solutions to various computational geometric problems on both networks. Computational geometry is just one area where the algorithms proposed here can be applied. Indeed, we believe that these algorithms are interesting and important in their own right and are fundamental to the design of solutions on the star and pancake networks to a host of other problems. © 1993 by John Wiley & Sons, Inc.

On locating path‐ or tree‐shaped facilities on networks

Abstract: The study of “optimally” locating on a network a single facility of a given total length in the form of a path or a tree was initiated by several authors. We extend these results to the problem of locating p (≥1) such facilities. We will consider “center”, “median”, “max eccentricity”, and “max distance sum” location type problems for p = 1 or p > 1, for general networks and for tree networks, whether a facility contains partial arcs or not, and whether a facility is path-shaped or tree-shaped. These cases lead to 64 problems. We will determine the algorithmic complexity of virtually all these problems. We conclude with a result that may be viewed as a generalization of the p-Median theorem. © 1993 by John Wiley & Sons, Inc.

Cycle prefix digraphs for symmetric interconnection networks

Abstract: Motivated by the study of large graphs with given degree and diameter, and the recent interest in the design of highly symmetric interconnection networks (e.g., the study of Cayley digraphs), we are led to the search for large vertex symmetric digraphs with given degree and diameter. The main result of this paper is the construction of a new class of vertex symmetric directed graphs, Γδ(D) (δ ≥ D) that have degree δ, diameter D, and (δ + 1)δ … (δ – D + 2) vertices. The graphs Γδ(D) are first found in the notation of Cayley coset digraphs. Then, we discover that they have a very simple representation in terms of sequences like the commonly studied networks such as the hypercube, de Bruijn graphs, and Kautz graphs. Based on the sequence representation, we give a simple shortest-path routing scheme. We also show that the average distance in our digraph Γδ(D) is very close to its diameter D. As a consequence, it follows that the natural routing scheme, which is even simpler than the shortest-path routing, is nearly optimal on an average basis. © 1993 by John Wiley & Sons, Inc.

Finding Minimum Cost to Time Ratio Cycles with Small Integral Transit Times

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An upper bound on the number of cliques in a graph

Abstract: Giving a partial solution to a conjecture of Balas and Yu [Networks19 (1989) 247–235], we prove that if the complement of a graph G on n vertices contains no set of t + 1 pairwise disjoint edges as an induced subgraph, then G has fewer than (n/2t)2t maximal complete subgraphs. © 1993 by John Wiley & Sons, Inc.

Doubly chordal graphs, steiner trees, and connected domination

Abstract: A new class of chordal graphs, containing the class of strongly chordal graphs, is introduced. It is shown that these graphs, called doubly chordal graphs, are related to acyclic hypergraphs and are recognizable in polynomial time. Furthermore, after proving that the Steiner tree and the connected domination problems are polynomially solvable for doubly chordal graphs, it is shown that both problems are NP-hard for a class of chordal graphs (called Helly chordal graphs) containing the class of doubly chordal graphs. © 1993 John Wiley & Sons, Inc.

A polynomial time solvable concave network flow problem

Abstract: We prove that the single-source uncapacitated (SSU) version of the concave cost network flow problem, when all arcs except one have linear cost, is in the class P of problems solvable in time polynomial in the problem input length. This contrasts the corresponding result without network constraints, in which the problem is known to be NP-hard [6]. © 1993 by John Wiley & Sons, Inc.

Finding all minimum‐size separating vertex sets in a graph

Abstract: We present a new algorithm based upon network flows for finding all minimum-size separating vertex sets in an undirected and unweighted graph. The sequential implementation of our algorithm runs in Θ(Mn + C) = O(2kn3) time, where M is the number of minimum-size separating vertex sets of the graph; n, the number of the vertices in the graph; m, the number of the edges in the graph; k, the connectivity of the graph, and C = kn min(k(m + n), A), where A is the complexity of the best maximum flow algorithm for unit networks. The parallel implementation runs either in O(k log n) deterministic time or in O(log2n) randomized time using Θ(;M2n2 + knNα) = O(4k(n6/k2)) processors on a PRAM, where Nα is the number of processors needed for parallel matrix multiplication in O(log n) time on PRAM. © 1993 by John Wiley & Sons, Inc.

A capacity scaling algorithm for the constrained maximum flow problem

Abstract: : The constrained maximum flow problem is to send the maximum possible flow from a source node 5 to a sink node tin 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 is the number of arcs, M is an upper bound on the largest element in the data, and s(n, m) is the time required to solve a shoflest path problem with nonnegative arc lengths. Our algorithms are modifications 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.

Shortest paths in stochastic networks with ARC lengths having discrete distributions

Abstract: In this work, we compute the distribution of L', the length of a shortest ( s , t ) path, in a directed network G with a source nodes and a sink node t and whose arc lengths are independent, nonnegative, integer valued random variables having finite support. We construct a discrete time Markov chain with a single absorbing state and associate costs with each transition such that the total cost incurred by this chain until absorption has the same distribution as does L'. We show that the transition probability matrix of this chain has an upper triangular structure and exploit this property to develop numerically stable algorithms for computing the distribution of L' and its moments. All the algorithms are recursive in nature and are illustrated by several examples. 0 7993byJohn Wiley & Sons, Inc.

A branch and bound algorithm for the capacitated minimum spanning tree problem

Abstract: Given an undirected graph G = (N, E) with a cost associated with each edge C: E → R+ and a demand associated with each node A: N → R+. A special node is designated as the center. The capacitated minimum spanning tree (CMST) problem is to find a minimum spanning tree for graph G such that the sum of demands on each branch stem from the center does not exceed a given capacity. The CMST problem has many applications in network design, centralized telecommunications, and vehicle routing. In this paper, we present a new formulation and a full optimization algorithm by branch and bound. The lower bounds are generated by Lagrangean relaxation with tightening constraints. Computational results based upon the methodology presented are shown. © 1993 by John Wiley & Sons, Inc.

Efficient parallel algorithms for bipartite permutation graphs

Abstract: In this paper, we further study the properties of bipartite permutation graphs. We give first efficient parallel algorithms for several problems on bipartite permutation graphs. These problems include transforming a bipartite graph into a strongly ordered one if it is also a permutation graph; testing isomorphism; finding a Hamiltonian path/cycle; solving a variant of the crossing number problem; and others. All these problems can be solved in O(log2n) time with O(n3) processors on a Common CRCW PRAM. We also show that the minimum fill-in problem for bipartite permutation graphs can be solved efficiently by a randomized parallel algorithm. © 1993 by John Wiley & Sons, Inc.

On the adjacency properties of Paley graphs

Abstract: In the application of graph theory to problems arising in network design, the requirements of the network can be expressed in terms of restrictions on the values of certain graph parameters such as connectivity, edge-connectivity, diameter, and independence number. In this paper, we focus on networks whose requirements translate into adjacency restrictions on the graph representing the network. More specifically, a graph G is said to have property P(m,n,k) if for any set of m + n distinct vertices there are at least k other vertices, each of which is adjacent to the first m vertices but not adjacent to any of the latter n vertices. The problem that arises is that of characterizing graphs having property P(m,n,k). In this paper, we present properties of graphs satisfying the adjacency property. In particular, for q 1(mod 4), a prime power, the Paley graph Gq of order q is the graph whose vertices are elements of the finite field q; two vertices are adjacent if and only if their difference is a quadratic residue. For any m, n, and k, we show that all sufficiently large Paley graphs satisfy P(m,n,k). © 1993 by John Wiley & Sons, Inc.

A Bayesian analysis of simulation algorithms for inference in belief networks

Abstract: A belief network is a graphical representation of the underlying probabilistic relationships in a complex system. Belief networks have been employed as a representation of uncertain relationships in computer-based diagnostic systems. These diagnostic systems provide assistance by assigning likelihoods to alternative explanatory hypotheses in response to a set of findings or observations. Approximation algorithms have been used to compute likelihoods of hypotheses in large networks. We analyze the performance of leading Monte Carlo approximation algorithms for computing posterior probabilities in belief networks. The analysis differs from earlier attempts to characterize the behavior of simulation algorithms in our explicit use of Bayesian statistics: We update a probability distribution over target probabilities of interest with information from randomized trials. For real ϵ, δ < 1 and for a probabilistic inference Pr[x|e], the output of an inference approximation algorithm in an (ϵ, δ)-estimate of Pr[x|e] if with probability at least 1 – δ the output is within relative error ϵ of Pr[x|e]. We construct a stopping rule for the number of simulations required by logic sampling, randomized approximation schemes, and likelihood weighting to provide (ϵ, δ)-estimates of Pr[x|e]. With Probability 1 – δ, the stopping rule is optimal in the sense that the algorithm performs the minimum number of required simulations. We prove that our stopping rules are insensitive to the prior probability distribution on Pr[x|e]. © 1993 by John Wiley & Sons, Inc.

Network transformations and bounding network reliability

Abstract: Three transformations on networks that reduce the all-terminal network reliability (probability of connectedness) of a network are shown not to increase any coefficient in one form of the reliability polynomial of the network. These transformations yield efficiently computable lower bounds on each coefficient of the reliability polynomial. A further transformation due to Lomonosov is shown not to decrease any coefficient in the reliability polynomial, leading to an efficiently computable upper bound on each coefficient. The resulting bounds on coefficients can, in turn, be used to obtain a substantial improvement on the Ball—Provan strategy for computing lower and upper bounds on the all-terminal reliability. © 1993 John Wiley & Sons, Inc.

Bounds for the general capacitated routing problem

Abstract: This paper presents heuristics that are based on a tour splitting of a general routing tour for solving the general capacitated routing problem (GCRP). This problem is a generalization of the vehicle routing problem (VRP) and the capacitated arc routing problem (CARP). For the VRP, heuristics that consist of an optimum partitioning of a TSP tour generated by Christofides are known and have a worst-case error of 7/2 − 3/q for even q, where q is the capacity of the vehicles. If we apply a partitioning to an optimum TSP tour, the worst-case error becomes 3 − 2/q for even q. We generalize these results to the GCRP and give also some lower bounds. © 1993 John Wiley & Sons, Inc.

A new method for constructing minimal broadcast networks

Abstract: Broadcast is the task of transmitting a message from any node in a network to all other nodes in the network. A minimal broadcast network (mbn) is a communication network in which a message can be broadcasted in minimum time regardless of the originator. In this article, a new method to construct such mbn's is presented. The new method improves the best known upper bounds on the minimum number of edges in mbn's for most cases. © 1993 by John Wiley & Sons, Inc.

Some minimum gossip graphs

Abstract: What is the minimum number of edges that a graph on n vertices must have to allow gossiping by bidirectional telephone calls within ⌈log2n⌉ rounds? Besides general estimates, this question is finally answered for integers of the form n = 2p − 2 (p ≥ 3) or n = 2p − 4 (p ≥ 6), and for n = 10, 12. Moreover, for n = 14, we prove the uniqueness of such a graph. © 1993 by John Wiley & Sons, Inc.

A differential game model of Nash equilibrium on a congested traffic network

Abstract: This paper considers the problem of the competition among a finite number of players who must transport the fixed volume of traffic on a simple network over a prescribed planning horizon. Each player attempts to minimize his total transportation cost by making simultaneous decisions of departure time, route, and flow rate over time. The problem is modeled as a N-person nonzero-sum differential game. Two solution concepts are applied: [1] the open-loop Nash equilibrium solution and [2] the feedback Nash equilibrium solution. Optimality conditions are derived and given an economic interpretation as a dynamic game theoretic generalization of Wardrop's second principle. Future extensions of the model are also discussed. The model promises potential applications to Intelligent Vehicle Highway Systems (IVHS) and air traffic control systems. © 1993 by John Wiley & Sons, Inc.

On stochastic spanning tree problem

Abstract: This paper considers a generalized version of the stochastic spanning tree problem in which edge costs are random variables and the objective is to find a spectrum of optimal spanning trees satisfying a certain chance constraint whose right-hand side also is treated as a decision variable. A special case of this problem with fixed right-hand side has been solved polynomially using a parameteric approach. Also, the same parametric method without increasing the complexity order has been extended to include the right-hand side also as a decision variable. In this paper, two different methods are given for solving the generalized problem. First, a different parametric method better than the earlier one is given. Then, a method that makes use of the efficient extreme points of the convex hull of the mappings of all the spanning trees in a bicriteria spanning tree problem is presented. But it is shown that in the worst case the bicriteria method is superior. © 1993 by John Wiley & Sons, Inc.

Undirected loop networks

Abstract: Let M, d, and k be positive integers. We say that M is feasible with respect to d and k if there exists a set A = {0, ± 1, ±a2 …, ±ak} such that the Cayley graph associated with Z/(M) and A has diameter less than or equal to d. Such a Cayley graph is a model for undirected loop communication networks. Denote M(d, k) the maximal feasible number M with respect to d and k. In the paper, an explicit formula for M(d, 2) is obtained. Also, when k ≧ 3, a lower bound for M(d, k) is established. © 1993 by John Wiley & Sons, Inc.

Linear-time Algorithms for the 2-connected Steiner Subgraph Problem On Special Classes of Graphs

Abstract: The 2-connected Steiner subgraph problem is that of finding a minimum-weight 2-connected subgraph that spans a subset of distinguished vertices. This paper presents linear-time algorithms for solving the 2-connected Steiner subgraph problem on two special classes of graphs, W4-free graphs and Halin graphs. Although different in detail, the algorithms adopt a common strategy exploiting known decompositions. As a special case, the algorithms also solve the Traveling Salesman Problem on W4-free graphs and Halin graphs.