## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

The (edge) forwarding index of a graph is the minimum, over all possible routings of all the demands, of the maximum load of an edge. This metric is of a great interest since it captures the notion of global congestion in a precise way: the lesser the forwarding-index, the lesser the congestion. In this paper, we study the following design question: Given a number e of edges and a number n of vertices, what is the least congested graph that we can construct? and what forwarding-index can we achieve? Our problem has some distant similarities with the well-known \((\varDelta ,D)\) problem, and we sometimes build upon results obtained on it. The goal of this paper is to study how to build graphs with low forwarding indices and to understand how the number of edges impacts the forwarding index. We answer here these questions for different families of graphs: general graphs, graphs with bounded degree, sparse graphs with a small number of edges by providing constructions, most of them asymptotically optimal. For instance, we provide an asymptotically optimal construction for \((n,n+k)\) cubic graphs - its forwarding index is \(\sim \frac{n^2}{3k} \log _2(k)\). Our results allow to understand how the forwarding-index drops when edges are added to a graph and also to determine what is the best (i.e. least congested) structure with e edges. Doing so, we partially answer the practical problem that initially motivated our work: If an operator wants to power only e links of its network, in order to reduce the energy consumption (or wiring cost) of its networks, what should be those links and what performance can be expected?

To read the full-text of this research,

you can request a copy directly from the authors.

ResearchGate has not been able to resolve any citations for this publication.

The (edge) forwarding index of a graph is the minimum, over all possible routings of all the demands, of the maximum load of an edge. This metric is of a great interest since it captures the notion of global congestion in a precise way: the lesser the forwarding-index, the lesser the congestion. In this paper, we study the following design question: Given a number e of edges and a number n of vertices, what is the least congested graph that we can construct? and what forwarding-index can we achieve? Our problem has some distant similarities with the well-known (∆,D) problem, and we sometimes build upon results obtained on it. The goal of this paper is to study how to build graphs with low forwarding indices and to understand how the number of edges impacts the forwarding index. We answer here these questions for different families of graphs: general graphs, graphs with bounded degree, sparse graphs with a small number of edges by providing constructions, most of them asymptotically optimal. Hence, our results allow to understand how the forwarding-index drops when edges are added to a graph and also to determine what is the best (i.e least congested) structure with e edges. Doing so, we partially answer the practical problem that initially motivated our work: If an operator wants to power only e links of its network, in order to reduce the energy consumption (or wiring cost) of its networks, what should be those links and what performance can be expected?

A routing R of a connected graph G is a collection that contains simple paths connecting every ordered pair of vertices in G. The edge-forwarding index with respect to R (or simply the forwarding index with respect to R) π(G, R) of G is the maximum number of paths in R passing through any edge of G. The forwarding index π(G) of G is the minimum π(G, R) over all routings R’s of G. This parameter has been studied for different graph classes [14], [1], [7], [5]. Motivated by energy efficiency, we look, for different numbers of edges, at the best spanning graphs of a square grid, namely those with a low forwarding index.

In order to optimize energy efficiency, network operators try to switch off as many network devices as possible. Recently,
there is a trend to introduce content caches as an inherent capacity of network equipment, with the objective of improving
the efficiency of content distribution and reducing network congestion. In this work, we study the impact of using in-network
caches and content delivery network (CDN) cooperation on an energy efficient routing. We formulate this problem as Energy
Efficient Content Distribution; we propose an integer linear program and a heuristic algorithm to solve it. The objective
of this problem is to find a feasible routing, so that the total energy consumption of the network is minimized while the
constraints given by the demands and the link capacity are satisfied. We exhibit for which range of parameters (size of caches,
popularity of content, demand intensity, etc.) it is useful to use caches. Experimental results show that by placing a cache
on each backbone router to store the most popular content, along with choosing well the best content provider server for each
demand to a CDN, we can save about 20% of power on average in all the backbone networks considered.

According to several studies, telecom operators account for a not negligible power consumption, and several initiatives are being put into place to reduce the power consumption of the ICT sector in general. To this goal, we propose a novel ap- proach to switch off some portions of the UMTS core network while still guaranteeing full connectivity and maximum link utilization. After showing that the problem falls in the class of capacitated multi-commodity flow problems, and therefore it is NP-complete, we propose some heuristic algorithms to solve it. Results show that it is possible to reduce the number of links and nodes currently used by up to 30% and 50% respectively during off-peak hours while offering the same service quality. I I NTRODUCTION Power consumption in general, and of ITC technologies in particular, has become a key issue during the last few years. The ratio of power demand versus power resources is con- stantly growing, and energy costs are increasing at a constant rate. The electricity cost jumped up of about 35% in Italy dur- ing the period 2004-2007 (4). Moreover, Green House Gases (GHG) emissions have a negative impact on the world climate change (8), and people are becoming more conscious about problems that will arise in the near future due to this. According to a number of studies, ICT alone is responsible for a percentage which varies from 2% to 10% of the world power consumption (7), due to the ever increasing diffusion of electronic devices. In this scenario, the power consumption of telecommunication networks is not negligible. For example, Telecom Italia consumes more than 2 TWh of energy, repre- senting the second national user (6) after the national railway operator. More recently, telecom operators have begun to adopt equip- ments that are able to be enter into standby mode (1). In this paper, we consider the packet switching domain of a typical UMTS architecture. Given the network topology and a traffic demand from the Radio Access Network, we evaluate the possibility to turn off some elements (nodes and links) un- der connectivity and Quality of Service (QoS) constraints. The goal is to minimize the total power consumption of the core network, in which usually resource over-provisioning is large. We investigate some simple optimization algorithms. In partic- ular, we selectively power off nodes and links of the topology following different strategies. Results show that it is possible to reduce the percentage of nodes and links actually powered on up to 30% and 50% respectively, while guaranteeing that the resource utilization is still below a given threshold. e.g., 50%. The paper is organized as follows: Sect. II presents the sce- nario and the problem formulation; Sect. III describes the im- plemented heuristics, and results are presented in Sect. IV. Fi- nally, conclusions are drawn in Sect. V.

In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect the metric of the (possibly weighted) graph. Given a graph G we map its vertices to a normed space in an attempt to (i) keep down the dimension of the host space, and (ii) guarantee a small distortion, i.e., make sure that distances between vertices in G closely match the distances between their geometric images. In this paper we develop efficient algorithms for embedding graphs low-dimensionally with a small distortion. Further algorithmic applications include: • A simple, unified approach to a number of problems on multicommodity flows, including the Leighton-Rao Theorem [37] and some of its extensions. We solve an open question in this area, showing that the max-flow vs. min-cut gap in the k-commodities problem is O(log k). Our new deterministic polynomial-time algorithm finds a (nearly tight) cut meeting this bound. • For graphs embeddable in low-dimensional spaces with a small distortion, we can find low-diameter decompositions (in the sense of [7] and [43]). The parameters of the decomposition depend only on the dimension and the distortion and not on the size of the graph. • In graphs embedded this way, small balanced separators can be found efficiently. Given faithful low-dimensional representations of statistical data, it is possible to obtain meaningful and efficient clustering. This is one of the most basic tasks in pattern-recognition. For the (mostly heuristic) methods used in the practice of pattern-recognition, see [20], especially chapter 6. Our studies of multicommodity flows also imply that every embedding of (the metric of) an n-vertex, constant-degree expander into a Euclidean space (of any dimension) has distortion Ω(log n). This result is tight, and closes a gap left open by Bourgain [12].

As community concerns about global energy consumption grow, the power consumption of the Internet is becoming an issue of increasing importance. In this paper, we present a network-based model of power consumption in optical IP networks and use this model to estimate the energy consumption of the Internet. The model includes the core, metro and edge, access and video distribution networks, and takes into account energy consumption in switching and transmission equipment. We include a number of access technologies, including digital subscriber line with ADSL2+, fiber to the home using passive optical networks, fiber to the node combined with very high-speed digital subscriber line and point-to-point optical systems. In addition to estimating the power consumption of today's Internet, we make predictions of power consumption in a future higher capacity Internet using estimates of improvements in efficiency in coming generations of network equipment. We estimate that the Internet currently consumes about 0.4% of electricity consumption in broadband-enabled countries. While the energy efficiency of network equipment will improve, and savings can be made by employing optical bypass and multicast, the power consumption of the Internet could approach 1% of electricity consumption as access rates increase. The energy consumption per bit of data on the Internet is around 75 mu J at low access rates and decreases to around 2-4 mu J at an access rate of 100 Mb/s.

A routing $R$ of a given connected graph $G$ of order $n$ is a collection of
$n(n-1)$ simple paths connecting every ordered pair of vertices of $G$. The
vertex-forwarding index $\xi(G,R)$ of $G$ with respect to $R$ is defined as the
maximum number of paths in $R$ passing through any vertex of $G$. The
vertex-forwarding index $\xi(G)$ of $G$ is defined as the minimum $\xi(G,R)$
over all routing $R$'s of $G$. Similarly, the edge-forwarding index $ \pi(G,R)$
of $G$ with respect to $R$ is the maximum number of paths in $R$ passing
through any edge of $G$. The edge-forwarding index $\pi(G)$ of $G$ is the
minimum $\pi(G,R)$ over all routing $R$'s of $G$. The vertex-forwarding index
or the edge-forwarding index corresponds to the maximum load of the graph.
Therefore, it is important to find routings minimizing these indices and thus
has received much research attention in the past ten years and more. In this
paper we survey some known results on these forwarding indices, further
research problems and several conjectures.

A network is defined as an undirected graph and a routing which consists of a collection of simple paths connecting every pair of vertices in the graph. The forwarding index of a network is the maximum number of paths passing through any vertex in the graph. Thus it corresponds to the maximum amount of forwarding done by any node in a communication network with a fixed routing. For a given number of vertices, each having a given degree constraint, the problem of finding networks that minimize the forwarding index is considered. Forwarding indexes are calculated for cube networks and generalized de Bruijn networks. General bounds are derived which show that de Bruijn networks are asymptotically optimal. Finally, efficient techniques for building large networks with small forwarding indexes out of given component networks are presented and analyzed.

Several studies exhibit that the traffic load of the routers only has a small influence on their energy consumption. Hence, the power consumption in networks is strongly related to the number of active network elements, such as interfaces, line cards, base chassis,... The goal thus is to find a routing that minimizes the (weighted) number of active network elements used when routing. In this paper, we consider a simplified architecture where a connection between two routers is represented as a link joining two network interfaces. When a connection is not used, both network interfaces can be turned off. Therefore, in order to reduce power consumption, the goal is to find the routing that minimizes the number of used links while satisfying all the demands. We first define formally the problem and we model it as an integer linear program. Then, we prove that this problem is not in APX, that is there is no polynomial-time constant-factor approximation algorithm. We propose a heuristic algorithm for this problem and we also prove some negative results about basic greedy and probabilistic algorithms. Thus we present a study on specific topologies, such as trees, grids and complete graphs, that provide bounds and results useful for real topologies. We then exhibit the gain in terms of number of network interfaces (leading to a global reduction of approximately 33 MWh for a medium-sized backbone network) for a set of existing network topologies: we see that for almost all topologies more than one third of the network interfaces can be spared for usual ranges of operation. Finally, we discuss the impact of energy efficient routing on the stretch factor and on fault tolerance.

A routing R of a connected graph G is a collection that contains simple paths connecting every ordered pair of vertices in G. The edge-forwarding index with respect to R (or simply the forwarding index with respect to R) π(G,R) of G is the maximum number of paths in R passing through any edge of G. The forwarding index π(G) of G is the minimum π(G,R) over all routings R's of G. This parameter has been studied for different graph classes (Xu and Xu, 2012; Bouabdallah and Sotteau, 1993; Fernandez de la Vega and Gordone, 1992; de la Vega and Manoussakis, 1992). Motivated by energy efficiency, we look, for different numbers of edges, at the best spanning graphs of a square grid, namely those with a low forwarding index.

In order to optimize energy efficiency, network operators try to switch off as many network devices as possible. Recently, there is a trend to introduce content caches as an inherent capacity of network equipment, with the objective of improving the efficiency of content distribution and reducing network congestion. In this work, we study the impact of using in-network caches and content delivery network (CDN) cooperation on an energy efficient routing.We formulate this problem as Energy Efficient Content Distribution; we propose an integer linear program and a heuristic algorithm to solve it. The objective of this problem is to find a feasible routing, so that the total energy consumption of the network is minimized while the constraints given by the demands and the link capacity are satisfied. We exhibit for which range of parameters (size of caches, popularity of content, demand intensity, etc.) it is useful to use caches. Experimental results show that by placing a cache on each backbone router to store the most popular content, along with choosing well the best content provider server for each demand to a CDN, we can save about 20% of power on average in all the backbone networks considered.

The maximum concurrent flow problem (MCFP) is a multicommodity flow problem in which every pair of entities can send and receive flow concurrently. The ratio of the flow supplied between a pair of entities to the predefined demand for that pair is called throughput and must be the same for all pairs of entities for a concurrent flow. The MCFP objective is to maximize the throughput, subject to the capacity constraints. We develop a fully polynomial-time approximation scheme for the MCFP for the case of arbitrary demands and uniform capacity. Computational results are presented. It is shown that the problem of associating costs (distances) to the edges so as to maximize the minimum-cost of routing the concurrent flow is the dual of the MCFP. A path-cut type duality theorem to expose the combinatorial structure of the MCFP is also derived. Our duality theorems are proved constructively for arbitrary demands and uniform capacity using the algorithm. Applications include packet-switched networks [1, 4, 8], and cluster analysis [16].

A routing R of a graph G is a set of n(n − 1) elementary paths R(u, v) specified for all ordered pairs (u, v) of vertices of G. The vertex-forwarding index ξ(G) of G, is defined by
Where ξ(G, R) is the maximum number of paths of the routing R passing through any vertex of G and the minimum is taken over all the routings of G. Let Gp denote the random graph on n vertices with edge probability p and let m = np. It is proved among other things that, under natural growth conditions on the function p = p(n), the ratio
Tends to 1 in probability as n tends to infinity.

For a given graph G of order n, a routing R is a set of n(n − 1) elementary paths specified for every ordered pair of vertices in G. The edge forwarding index of a network (G,R), denoted π(G,R) is the maximum number of paths of R going through any edge e of G. The edge forwarding index of G, denoted π(G), is the minimum of π(G,R) taken over all the possible routings R of G. Given n ≤ 15 and Δ ≤ n − 1 we determine πΔ,n, the minimum of π(G) taken over all graphs G of order n with maximum degree at most Δ. This is known as the edge forwarding index problem. © 1993 by John Wiley & Sons, Inc.

This article describes the Survivable Network Design Library (SNDlib), a data library for fixed telecommunication network design available at http://sndlib.zib.de. In the current version 1.0, the library contains data related to 22 networks which, combined with a set of selected planning parameters, leads to 830 network design problem instances. In this article, we discuss the data concepts of SNDlib and describe a mathematical model for each design problem considered in the library. We also provide information on characteristic features and the origin of the SNDlib problem instances. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

Energy efficient communication devices are essential to minimize the operational cost of future networks and to reduce the negative effects of global warming. In this paper we propose a novel energy reduction approach on network level that takes load-dependent energy consumption information of communication equipment into account. Case study calculation results show that energy savings of more than 35% and with it operational cost can be saved by applying energy profile aware routing.

A network (G,R) consists in a given undirected graph G of order n and a routing R, that is a collection of n(n-1) simple paths connecting every ordered pair of vertices of G. Chung, Coffman, Reiman and Simon defined the forwarding index ξ(G,R) of a network (G,R) as the maximum number of paths of R passing through any vertex of G. Similarly we define the edge-forwarding index of a network (G,R) as the maximum number of paths of R passing through any edge of G. These parameters might be of interest in different applications concerning communication networks. The forwarding (resp. edge-forwarding) index corresponds to the maximum amount of forwarding done by any node (resp. edge). The edge-forwarding index also corresponds to the maximum load of the network. Therefore it is of interest, for a given graph, to find routings minimizing these indices and we shall define the forwarding (edge-forwarding) index of a graph as the minimum taken over all possible indices of the possible networks.In this paper we give bounds on these forwarding indices, in particular as a function of the connectivity of the graph, and calculate them for products of graphs and for some specific graphs.

Expanding parameters of graphs (magnification constant, edge and vertex cutset expansion) are related by very simple inequalities to forwarding parameters (edge and vertex forwarding indices). This shows that certain graphs have eccentricity close to the diameter. Connections between the forwarding indices and algebraic parameters like the smallest eigenvalue of the Laplacian or the genus of the graph are made. Graphs with unknown spectrum (de Bruijn, Kautz) are shown to be reasonable expanding by purely combinatorial arguments. Conversely, near-optimal routings in these graphs yield tight bounds on the spectrum.

Answering some questions of Heydemann, Meyer, Opatrny and Sotteau [4], we give upper bounds for the forwarding index of graphs of order n and given connectivity k. We first prove that in a k-connected (respectively k-edge connected) graph with n substantially larger than k, the edge-forwarding index of shortest paths is no more than n2/2-(k-1)n+3(k-1)2/8 (respectively [n2/2]-n-2(k-1)2). We next prove that for a k-connected graph the vertex- and edge-forwarding indices are no more than (n-1[(n-k-1)/k] and n[(n-k-1)/k], respectively. Related conjectures are proposed.

In a given network with n vertices, a routing is defined as a set of n(n — 1) paths, one path connecting each ordered pair of vertices. The load of a vertex is the number of paths going through it. The forwarding index of the network is the minimum of the largest load taken over all routings. We give upper bounds on the forwarding index in k-connected digraphs and in digraphs with half-degrees at least k. Related conjectures are proposed.

In this paper, we establish max-flow min-cut theorems for several important classes of multicommodity flow problems. In particular, we show that for any n-node multicommodity flow problem with uniform demands, the max-flow for the problem is within an O(log n) factor of the upper bound implied by the min-cut. The result (which is existentially optimal) establishes an important analogue of the famous 1-commodity max-flow min-cut theorem for problems with multiple commodities. The result also has substantial applications to the field of approximation algorithms. For example, we use the flow result to design the first polynomial-time (polylog n-times-optimal) approximation algorithms for well-known NP-hard optimization problems such as graph partitioning, min-cut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. Applications of the flow results to path routing problems, network reconfiguration, communication in distributed networks, scientific computing and rapidly mixing Markov chains are also described in the paper. Categories and Subject Descriptors: F.2.2 (Analysis of Algorithms and Problem Complexity):

The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds { called Moore bounds { for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maxi- mum possible number of vertices, given the other two parameters, and thus attack- ing the degree/diameter problem 'from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem 'from below'. This sur- vey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.

The stability of a queueing network with interdependent servers is
considered. The dependency among the servers is described by the
definition of their subsets that can be activated simultaneously.
Multihop radio networks provide a motivation for the consideration of
this system. The problem of scheduling the server activation under the
constraints imposed by the dependency among servers is studied. The
performance criterion of a scheduling policy is its throughput that is
characterized by its stability region, that is, the set of vectors of
arrival and service rates for which the system is stable. A policy is
obtained which is optimal in the sense that its stability region is a
superset of the stability region of every other scheduling policy, and
this stability region is characterized. The behavior of the network is
studied for arrival rates that lie outside the stability region.
Implications of the results in certain types of concurrent database and
parallel processing systems are discussed

The paper is concerned with tools for the quantitative analysis of finite Markov chains whose states are combinatorial structures. Chains of this kind have algorithmic applications in many areas, including random sampling, approximate counting, statistical physics and combinatorial optimisation. The efficiency of the resulting algorithms depends crucially on the mixing rate of the chain, i.e. , the time taken for it to reach its stationary or equilibrium distribution.
The paper presents a new upper bound on the mixing rate, based on the solution to a multicommodity flow problem in the Markov chain viewed as a graph. The bound gives sharper estimates for the mixing rate of several important complex Markov chains. As a result, improved bounds are obtained for the runtimes of randomised approximation algorithms for various problems, including computing the permanent of a 0–1 matrix, counting matchings in graphs, and computing the partition function of a ferromagnetic Ising system. Moreover, solutions to the multicommodity flow problem are shown to capture the mixing rate quite closely: thus, under fairly general conditions, a Markov chain is rapidly mixing if and only if it supports a flow of low cost.

In the last decade important relations between Laplace eigenvalues and eigenvectors

Small cubic graphs, flinders Univ projet

- M Meringer