Spectral bounds for the betweenness of a graph

In this paper we find spectral bounds (Laplacian matrix) for the vertex and the edge betweenness of a graph. We also relate the edge betweenness with the isoperimetric number and the edge forwarding and edge expansion indices of the graph allowing a new upper bound on its diameter. The results are of interest as they can be used in the study of communication properties of real networks, in particular for dynamical processes taking place on them (broadcasting, network synchronization, virus spreading, etc.).     

The time complexity of deterministic broadcast radio networks

Bar-Yehuda, Goldreich and Itai studied both the probabilistic and the deterministic time complexity of broadcast in a multihop network. They show that in the deterministic case the number of time-slots needed for successful broadcast is lower bounded by the numbers of moves of two associated games. They also obtain lower bounds for the numbers of moves in these games. In this paper we determine the exact numbers of moves and time-slots needed in the games and in broadcasting.     

Optimizing cost flows by edge cost and capacity upgrade

We examine a network upgrade problem for cost flows. A budget can be distributed among the arcs of the network. An investment on each single arc can be used either to decrease the arc flow cost, or to increase the arc capacity, or both. The goal is to maximize the flow through the network while not exceeding bounds on the budget and on the total flow cost.

The problems are NP-hard even on series-parallel graphs. We provide an approximation algorithm on series-parallel graphs which, for arbitrary δ,>0, produces a solution which exceeds the bounds on the budget and the flow cost by factors of at most 1+δ and 1+, respectively, while the amount of flow is at least that of an optimum solution. The running time of the algorithm is polynomial in the input size and 1/(δ). In addition we give an approximation algorithm on general graphs applicable to problem instances with small arc capacities.     

Broadcasting from multiple originators

We begin an investigation of broadcasting from multiple originators, a variant of broadcasting in which any k vertices may be the originators of a message in a network of n vertices. The requirement is that the message be distributed to all n vertices in minimum time. A minimumk-originator broadcast graph is a graph on n vertices with the fewest edges such that any subset of k vertices can broadcast in minimum time. B_k(n) is the number of edges in such a graph. In this paper, we present asymptotic upper and lower bounds on B_k(n). We also present an exact result for the case when . We also give an upper bound on the number of edges in a relaxed version of this problem in which one additional time unit is allowed for the broadcast.     

Analysis of an open tandem queueing network with population constraint and constant service times 1

We consider an open tandem queueing network with population constraint and constant service times. The total number of customers that may be present in the network can not exceed a given value K. Customers arriving at the queueing network when there are more than K customers are forced to wait in an external queue. The arrival process to the queueing network is assumed to be arbitrary. We show that this queueing network can be transformed into a simple network involving only two nodes. Using this simple network, we obtain an upper and lower bound on the mean waiting time. These bounds can be easily calculated. Validations against simulation data establish the tightness of these bounds.     

Towards an Erlang formula for multiclass networks

Consider a multiclass stochastic network with state-dependent service rates and arrival rates describing bandwidth-sharing mechanisms as well as admission control and/or load balancing schemes. Given Poisson arrival and exponential service requirements, the number of customers in the network evolves as a multi-dimensional birth-and-death process on a finite subset of ℕ ^k . We assume that the death (i.e., service) rates and the birth rates depending on the whole state of the system satisfy a local balance condition. This makes the resulting network a Whittle network, and the stochastic process describing the state of the network is reversible with an explicit stationary distribution that is in fact insensitive to the service time distribution. Given routing constraints, we are interested in the optimal performance of such networks. We first construct bounds for generic performance criteria that can be evaluated using recursive procedures, these bounds being attained in the case of a unique arrival process. We then study the case of several arrival processes, focusing in particular on the instance with admission control only. Building on convexity properties, we characterize the optimal policy, and give criteria on the service rates for which our bounds are again attained.     

A dynamic bounding algorithm for approximating multi-state two-terminal reliability

This paper modifies Jane and Laih’s (2008) exact and direct algorithm to provide sequences of upper bounds and lower bounds that converge to the NP-hard multi-state two-terminal reliability. Advantages of the modified algorithm include (1) it does not require a priori the lower and/or upper boundary points of the network, (2) it derives a series of increasing lower bounds and a series of decreasing upper bounds simultaneously, guaranteed to enclose the exact reliability value, and (3) trade-off between accuracy and execution time can be made to ensure an exact difference between the upper and lower bounds within an acceptable time. Examples are analyzed to illustrate the bounding algorithm, and to compare the bounding algorithm with existing algorithms. Computational experiments on a large network are conducted to realize the performance of the bounding algorithm.     

Restricted recourse strategies for bounding the expected network recourse function

This paper presents bounds for the expected recourse function for stochastic programs with network recourse. Cyclic recourse, a concept introduced by Wallace [18], allows the approximation of the recourse problem by restricting the optimal flows on a set of cycles and by augmenting the original network to induce separability. We introduce a new procedure that uses again a set of cycles but does not approximate the problem; instead it solves it heuristically without altering the original network or requiring separability. The method produces tighter bounds and is computationally feasible for large networks. Numerical experiments with selected networks illustrate the effectiveness of the approach.     

Telecommunication Link Restoration Planning with Multiple Facility Types

To ensure uninterrupted service, telecommunication networks contain excess (spare) capacity for rerouting (restoring) traffic in the event of a link failure. We study the NP-hard capacity planning problem of economically installing spare capacity on a network to permit link restoration of steady-state traffic. We present a planning model that incorporates multiple facility types, and develop optimization-based heuristic solution methods based on solving a linear programming relaxation and minimum cost network flow subproblems. We establish bounds on the performance of the algorithms, and discuss problem instances that nearly achieve these worst-case bounds. In tests on three real-world problems and numerous randomly-generated problems containing up to 50 nodes and 150 edges, the heuristics provide good solutions (often within 0.5% of optimality) to problems with single facility type, in equivalent or less time than methods from the literature. For multi-facility problems, the gap between our heuristic solution values and the linear programming bounds are larger. However, for small graphs, we show that the optimal linear programming value does not provide a tight bound on the optimal integer value, and our heuristic solutions are closer to optimality than implied by the gaps.     

On the computational complexity of continuous routing

We study the problem of routing and broadcasting messages in a network, in which messages are generated at processors at arbitrary times and each message must reach its destination by a specific deadline. We present distributed and global routing algorithms for some restricted continuous routing problems on arrays of processors. We show that distributed algorithms are unlikely to exist in more general situations by giving an NP-hardness proof for their corresponding feasibility problem; i.e., the problem of determining whether all messages can be routed without violating the constraints of the network. We also present a distributed algorithm for the continuous broadcasting problem.     

Broadcasting in geometric radio networks

We consider deterministic broadcasting in geometric radio networks (GRN) whose nodes know only a limited part of the network. Nodes of a GRN are situated in the plane and each of them is equipped with a transmitter of some range r. A signal from this node can reach all nodes at distance at most r from it but if a node u is situated within the range of two nodes transmitting simultaneously, then a collision occurs at u and u cannot get any message. Each node knows the part of the network within knowledge radius s from it, i.e., it knows the positions, labels and ranges of all nodes at distance at most s.The aim of this paper is to study the impact of knowledge radius s on the time of deterministic broadcasting in a GRN with n nodes and eccentricity D of the source. Our results show sharp contrasts between the efficiency of broadcasting in geometric radio networks as compared to broadcasting in arbitrary graphs. They also show quantitatively the impact of various types of knowledge available to nodes on broadcasting time in GRN. Efficiency of broadcasting is influenced by knowledge radius, knowledge of individual positions when knowledge radius is zero, and awareness of collisions.     

Analysis of a three node queueing network

The three node Jackson queueing network is the simplest acyclic network in which in equilibrium the sojourn times of a customer at each of the nodes are dependent. We show that assuming the individual sojourn times are independent provides a good approximation to the total sojourn time. This is done by simulating the network and showing that the sojourn times generally pass a Kolmogorov-Smirnov test as having come from the approximating distribution. Since the sum of dependent random variables may have the same distribution as the sum of independent random variables with the same marginal distributions, it is conceivable that our approximation is exact. However, we numerically compute upper and lower bounds for the distribution of the total sojourn time; these bounds are so close that the approximating distribution lies outside of the bounds. Thus, the bounds are accurate enough to distinguish between the two distributions even though the Kolmogorov-Smirnov test generally cannot.     

Computation of lower bounds on the network cost in location problems subject to distance constraints

Methods for the computation of lower bounds on the cost of the connecting network for the continuous and discrete variants of the problem of location of interconnected objects subject to minimal or maximal distances between them are proposed. For the continuous variant, the bound is found by solving a linear programming problem. For the discrete variant, an assignment problem with a rectangular matrix containing forbidden entries is constructed. An application of the assignment problem for locating objects of various sizes is described.     

Tight bounds for popping algorithms

We sharpen run‐time analysis for algorithms under the partial rejection sampling framework. Our method yields improved bounds for: the cluster‐popping algorithm for approximating all‐terminal network reliability; the cycle‐popping algorithm for sampling rooted spanning trees; and the sink‐popping algorithm for sampling sink‐free orientations. In all three applications, our bounds are not only tight in order, but also optimal in30 constants.     

On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds

We develop tight bounds and a fast parallel algorithm to compute the Markov renewal kernel. Knowledge of the kernel allows us to solve Markov renewal equations numerically to study non-steady state behavior in a finite state Markov renewal process. Computational error and numerical stability for computing the bounds in parallel are discussed using well-known results from numerical analysis. We use our algorithm and computed bounds to study the expected number of departures as a function of time for a two node overflow queueing network.     

Accuracy and Stability of the Null Space Method for Solving the Equality Constrained Least Squares Problem

The null space method is a standard method for solving the linear least squares problem subject to equality constraints (the LSE problem). We show that three variants of the method, including one used in LAPACK that is based on the generalized QR factorization, are numerically stable. We derive two perturbation bounds for the LSE problem: one of standard form that is not attainable, and a bound that yields the condition number of the LSE problem to within a small constant factor. By combining the backward error analysis and perturbation bounds we derive an approximate forward error bound suitable for practical computation. Numerical experiments are given to illustrate the sharpness of this bound.     

On Stochastic Bounds for Monotonic Processor Sharing Networks

We consider a network of processor sharing nodes with independent Poisson arrival processes. Nodes are coupled through their service capacity in that the speed of each node depends on the number of customers present at this and any other node. We assume the network is monotonic in the sense that removing a customer from any node increases the service rate of all customers. Under this assumption, we give stochastic bounds on the number of customers present at any node. We also identify limiting regimes that allow to test the tightness of these bounds. The bounds and the limiting regimes are insensitive to the service time distribution. We apply these results to a number of practically interesting systems, including the discriminatory processor sharing queue, the generalized processor sharing queue, and data networks whose resources are shared according to max–min fairness.     

Assignment and scheduling in parallel matrix factorization

We consider the problem of factoring a dense n×n matrix on a network consisting of P MIMD processors, with no shared memory, when the network is smaller than the number of elements in the matrix (P<n²). The specific example analyzed is a computational network that arises in computing the LU, QR, or Cholesky factorizations. We prove that if the nodes of the network are evenly distributed among processors and if computations are scheduled by a round-robin or a least-recently-executed scheduling algorithm, then optimal order of speedup is achieved. However, such speedup is not necessarily achieved for other scheduling algorithms or if the computation for the nodes is inappropriately split across processors, and we give examples of these phenomena. Lower bounds on execution time for the algorithm are established for two important node-assignment strategies.     

Interactive Multiple Criteria Decision Making based on preference driven Evolutionary Multiobjective Optimization with controllable accuracy

We present an approach to interactive Multiple Criteria Decision Making based on preference driven Evolutionary Multiobjective Optimization with controllable accuracy.The approach relies on formulae for lower and upper bounds on coordinates of the outcome of an arbitrary efficient variant corresponding to preference information expressed by the Decision Maker. In contrast to earlier works on that subject, here lower and upper bounds can be calculated and their accuracy controlled entirely within evolutionary computation framework. This is made possible by exploration of not only the region of feasible variants - a standard within evolutionary optimization, but also the region of infeasible variants, the latter to our best knowledge being a novel approach within Evolutionary Multiobjective Optimization.To illustrate how this concept can be applied to interactive Multiple Criteria Decision Making, two algorithms employing evolutionary computations are proposed and their usefulness demonstrated by a numerical example.     

Large Deviation Bounds for Single Class Queueing Networks and Their Calculation

We investigate large deviations for the behavior of single class queueing networks. The starting point is a sample large deviation principle on the path-space of network primitives describing the cumulative external arrivals, service time requirements and routing decisions. The behavior of the network, capturing the cumulative total arrivals, idle times and queue lengths, is characterized by a path-space fixed point equation containing the network primitives. The mapping from the network primitives to the set of fixed points is partially upper semicontinuous. This set-valued continuity allows us to derive large deviation bounds for the network behavior in the form of variational problems. The analysis is carried out on the doubly-infinite time axis R and can directly capture stationary and non-Markovian situations. By relaxing the fixed point equation the upper bounds and minimizing paths can be approximated with piecewise linear paths. For a class of typical rate functions we specify sequences of finite dimensional minimization problems which permit the calculation of large deviation rates and minimizing paths for the tail probabilities of queue lengths. We illustrate the approach with an example.