On Canonical Concurrent Flows, Crossing Number and Graph Expansion

The edge expansion of a graph is the minimum quotient of the number of edges in a cut and the size of the smaller one among the two node sets separated by the cut. Bounding the edge expansion from below is important for bounding the “mixing time” of a random walk on the graph from above. It has been conjectured by Mihail and Vazirani (see [9]) that the graph of every 0/1-polytope has edge expansion at least one. A proof of this (or even a weaker) conjecture would imply solutions of several long-standing open problems in the theory of randomized approximate counting. We present different techniques for bounding the edge expansion of a 0/1-polytope from below. By means of these tools we show that several classes of 0/1-polytopes indeed have graphs with edge expansion at least one. These classes include all 0/1-polytopes of dimension at most five, all simple 0/1-polytopes, all hypersimplices, all stable set polytopes, and all (perfect) matching polytopes.

The maximum multicommodity flow problem is a natural generalization of the maximum flow problem to route multiple distinct flows. Obtaining a 1-ε approximation to the multicommodity flow problem on graphs is a well-studied problem. In this paper we present an adaptation of recent advances in single-commodity flow algorithms to this problem. As the underlying linear systems in the electrical problems of multicommodity flow problems are no longer Laplacians, our approach is tailored to generate specialized systems which can be preconditioned and solved efficiently using Laplacians. Given an undirected graph with m edges and k commodities, we give algorithms that find 1-ε approximate solutions to the maximum concurrent flow problem and maximum weighted multicommodity flow problem in time O(m4/3poly(k,ε-1)).

In the concurrent multi-commodity flow problem, we are given a capacitated network G = (V,E) of switches V connected by links E, and a set of commodities \({\cal K} = \{(s_i,t_i,d_i)\}\). The objective is to maximize the minimum fraction λ of any demand d i that is routed from source s i to target t i . This problem has been studied extensively by the theoretical computer science community in the sequential model (e.g., [4]) and in distributed models (e.g., [2,3]). Solutions in the networking systems community also fall into these models (e.g., [1,6,5]), yet none of them use the state-of-the-art algorithms above. Why the gap between theory and practice? This work seeks to answer and resolve this question. We argue that existing theoretical models are ill-suited for real networks (§2) and propose a new distributed model that better captures their requirements (§3). We have developed optimal algorithms in this model for data center networks (§4); making these algorithms practical requires a novel use of programmable hardware switches. A solution for general networks poses an intriguing open problem.

Decomposition based approaches are recalled from primal and dual point of view. The possibility of building partially disaggregated reduced master problems is investigated. This extends the idea of aggregated-versus-disaggregated formulation to a gradual choice of alternative level of aggregation. Partial aggregation is applied to the linear multicommodity minimum cost flow problem. The possibility of having only partially aggregated bundles opens a wide range of alternatives with different trade-offs between the number of iterations and the required computation for solving it. This trade-off is explored for several sets of instances and the results are compared with the ones obtained by directly solving the natural node-arc formulation. An iterative solution process to the route assignment problem is proposed, based on the well-known Frank Wolfe algorithm. In order to provide a first feasible solution to the Frank Wolfe algorithm, a linear multicommodity min-cost flow problem is solved to optimality by using the decomposition techniques mentioned above. Solutions of this problem are useful for network orientation and design, especially in relation with public transportation systems as the Personal Rapid Transit. A single-commodity robust network design problem is addressed. In this, an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. A set of new instances that are computationally hard for the natural flow formulation are solved by means of a new heuristic algorithm. Finally, an efficient decomposition-based heuristic approach for a large scale stochastic unit commitment problem is presented. The addressed real-world stochastic problem employs at its core a deterministic unit commitment planning model developed by the California Independent System Operator (ISO).

We continue the study of random lifts of graphs initiated in [4]. Here we study the possibility of generating graphs with high edge expansion as random lifts. Along the way, we introduce the method of -nets into the study of random structures. This enables us to improve (slightly) the known bounds for the edge expansion of regular graphs.

A parallel processing system's most crucial part is a network interconnection that links its processors. The hypercube topology has exciting features, making it an excellent option for parallel processing applications. This paper presents two innovative configurations of interconnection networks based on fractal Sierpinski and a hypercube. These are the Sierpinski Triangle Topology (STT) and Sierpinski Carpet Topology (SCT). Compared to a hypercube, the Sierpinski Triangle topology (STT) noticed a significant decrease in the number of nodes and links as large networks grew. Hence, it considers a great way to reduce costs because it uses fewer nodes and links. The average distance is also shorter, which is better. Despite it has a smaller bisection width and a more degree than a hypercube by one. The Sierpinski Carpet Topology (SCT) has the advantage of having a higher bisection width than a hypercube. That is preferable because it places a lower restriction on the difficulty of parallel algorithms. In contrast, the drawback of this topology is that it has a diameter and average distance more than a hypercube.

Message passing equations yield a sharp percolation transition in finite graphs, as an artifact of the locally treelike approximation. For an arbitrary finite, connected, undirected graph we construct an infinite tree having the same local structural properties as this finite graph, when observed by a nonbacktracking walker. Formally excluding the boundary, this infinite tree is a generalization of the Bethe lattice. We indicate an infinite, locally treelike, random network whose local structure is exactly given by this infinite tree. Message passing equations for various cooperative models on this construction are the same as for the original finite graph, but here they provide the exact solutions of the corresponding cooperative problems. These solutions are good approximations to observables for the models on the original graph when it is sufficiently large and not strongly correlated. We show how to express these solutions in the critical region in terms of the principal eigenvector components of the nonbacktracking matrix. As representative examples we formulate the problems of the random and optimal destruction of a connected graph in terms of our construction, the nonbacktracking expansion. We analyze the limitations and the accuracy of the message passing algorithms for different classes of networks and compare the complexity of the message passing calculations to that of direct numerical simulations. Notably, in a range of important cases, simulations turn out to be more efficient computationally than the message passing.

Many transportation, communications, and logistics problems can be formulated as large multi-commodity flow problems, which are often solved using the traditional column generation method. During the application of the column generation, one needs to find initial solutions, so-called feasible columns in a first step. In order to determine these feasible columns and also solve dependent price problems within an acceptable time, this paper proposes several novel methods: Two methods by solving the global problem and two methods by finding the shortest and widest paths are proposed to compute the initial columns. Furthermore, we propose two methods by finding the shortest paths with commodities and sources (SPC and SPS) for solving price problems. Several well-known network instances are used as case studies in our evaluation to test the efficiency of each method. It is shown that the widest-path method is the best for finding initial columns. For solving the price problems, the preference of the two path-finding methods depend on the network structure.

The advancements in technology in this contemporary time require special computing devices to able to manage the increase of the computational power. However, the limitation of the sequential computing systems stands as an obstacle in fulfilling these requirements. Thus, the research community looking for a viable solution able to solve the grand computing problems. A parallel computing system is a collection of processing elements that cooperate and communicate to provide a fast solution for a problem. A new symmetric network named Shifted Peterson Network (SPN) has been proposed in this paper. The basic module (BM) of the suggested network is composed of ten nodes, and the BMs of SPN are connected through external nodes to construct an advanced level of this network. Furthermore, the connectivity of SPN is based on a shifting mechanism of the binary digits of each node. The architecture of SPN discussed in this paper, in addition, the static network performance parameters of this network has been evaluated and compared to popular conventional interconnection networks. The networks evaluated in terms of node degree, diameter, average distance, arc connectivity, bisection width, cost, wiring complexity, packing density, and message traffic density. SPN showed good results in almost all aspects compared to these networks. Therefore, SPN is a suitable choice for the massively parallel computer (MPC) systems.

In this paper, we have proposed a new hierarchical interconnection network called Tori-connected Flattened Butterfly Network (TFBN). It is a hierarchical interconnection network which is able to interconnect millions of nodes in the network to achieve high performance and cost-effective design for massively parallel computing system. TFBN consists of multiple basic modules of flattened butterfly network whereby the basic modules are connected using 2D-torus network. This paper focuses on the architecture of the TFBN and the evaluation of its static network performance such as degree, diameter, cost, average distance, arc connectivity, bisection width and wiring complexity. The comparison of performance shows better performance in terms of distance parameters as well as other parameters.

This paper presents a QoS aware fair routing and bandwidth management method for transport networks. We consider a variant of the maximum concurrent multicommodity flow problem where only paths of bounded length are allowed to carry flow. We show two possible approaches to the problem. First, we give a column generation method, where the Bellman-Ford algorithm provides the path of bounded length with minimum reduced cost. Then we present an ? - approximation scheme to the problem, applying the method of Garg and Konemann with new ideas for speedup. Finally the computational experience is presented.

Linial, London and Rabinovich [16] and Aumann and Rabani [3] proved that the min-cut max-flow ratio for general maximum concurrent flow problems (when there are k commodities) is O(logfe). Here we attempt to derive a more general theory of Steiner cut and flow problems, and we prove bounds that are poly-logarithmic in k for a much broader class of multicommodity flow and cut problems. Our structural results are motivated by the meta question: Suppose we are given a poly(log n) approximation algorithm for a flow or cut problem when can we give a poly(log k) approximation algorithm for a generalization of this problem to a Steiner cut or flow problem? Thus we require that these approximation guarantees be independent of the size of the graph, and only depend on the number of commodities (or the number of terminal nodes in a Steiner cut problem). For many natural applications (when k = n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o(1)</sup> ) this yields much stronger guarantees. We construct vertex-sparsifiers that approximately preserve the value of all terminal min-cuts. We prove such sparsifiers exist through zero-sum games and metric geometry, and we construct such sparsifiers through oblivious routing guarantees. These results let us reduce a broad class of multicommodity-type problems to a uniform case (on k nodes) at the cost of a loss of a poly (log k) in the approximation guarantee. We then give poly(log k) approximation algorithms for a number of problems for which such results were previously unknown, such as requirement cut, 1-multicut, oblivious 0-extension, and natural Steiner generalizations of oblivious routing, min-cut linear arrangement and minimum linear arrangement.

An optical interconnection architecture is proposed, which is based on optical ring and torus. The proposed optical interconnection network combines advantages of both ring and torus, e.g., simple node interface, constant node degree, better support for local communication and remote communication, wide bisection and good scalability etc. This architecture could be used to connect thousands of processing elements in massively parallel computer systems while maintaining a low latency and high bandwidth. The comparison with other popular networks is made, and the result shows that RTOIN (Ring based Torus Optical Interconnection Network) has pretty good node degree, network diameter, the number of links, bisection width and average message distance. Also, the result shows its excellent scalability, so it will probably be widely used in the area of parallel computing in future.

A one-dimensional mathematical model is developed for an unsteady incompressible laminar flow in a circular pipe subjected to a rapid change in the flow rate from an initial flow with flow rate, Qi, to a final flow with flow rate, Qf, in a step-like fashion at an arbitrary time, tc. The change in the flow rate may either be an increment, Qf &amp;gt; Qi, or a decrement, Qf &amp;lt; Qi. The change time, tc, may either belong to the initial flow remaining in a temporally developing state or temporally developed state. The developed model is solved using the Laplace transform method to deduce generalized analytical expressions for the flow characteristics, viz., velocity, pressure gradient, wall shear stress, and skin friction factor, CfRe, where Re is Reynolds number based on the cross-sectional area-averaged velocity and pipe radius. Exact solutions for λa=Qi/Qf=0 and λd=Qf/Qi=0 with tc≥tsi are available in the literature and the present generalized analytical solutions fill the remaining range of parameters, 0&amp;lt;λa&amp;lt;1 and 0&amp;lt;λd&amp;lt;1 with 0&amp;lt;tc&amp;lt;tsi and tc≥tsi, where tsi is the time at which the initial flow reaches the temporally developed state. Exact solutions for canonical pipe flow problems reported in the literature are deduced as subsets of the derived generalized solutions. The parametric study reveals the effects of varying λa or λd and tc on the quantities of practical importance, viz., τs and CfRe, where τs is the time required for the final flow to reach the temporally developed state.

Local correctability of expander codes