Optimization of the capacity of wireless mesh networks

This is a summary of the authors PhD thesis supervised by Hervé Rivano and defended on 29 October 2009 at the Université de Nice-Sophia Antipolis. The thesis is written in French and is available from . This work deals with the optimization of the capacity of wireless mesh networks, defined as the throughput offered to each flow. We develop optimization models integrating the cross-layer characteristics of radio communications. The joint routing and scheduling is studied and solved using column generation. A linear formulation focusing on the transport capacity available on the network cuts is derived. We prove the equivalence of the models, and adapt the resolution method into a cross line and column generation process. Thorough tests, a contention area located around the gateways which constraints the capacity is highlighted. These results are applied to a quantitative study of the effects of acknowledgments on the capacity. Finally, a stability study of a protocol routing a traffic injected arbitrarily is investigated.     

Optimization architecture for joint multi-path routing and scheduling in wireless mesh networks

In Wireless Mesh Networks (WMN), the optimal routing of data depends on the link capacities which are determined by link scheduling. The optimal performance of the network, therefore, can only be achieved by joint routing and scheduling optimization. Although the joint single-path routing and scheduling optimization problem has been extensively studied, its multi-path counterpart within wireless mesh networks has not yet been fully investigated. In this paper, we present an optimization architecture for joint multi-path QoS routing and the underlying wireless link scheduling in wireless mesh networks. By employing the contention matrix to represent the wireless link interference, we formulate a utility maximization problem for the joint multi-path routing and MAC scheduling and resolve it using the primal–dual method. Since the multi-path routing usually results in the non-strict concavity of the primal objective function, we first introduce the Proximal Optimization Algorithm to get around such difficulty. We then propose an algorithm to solve the routing subproblem and the scheduling subproblem via the dual decomposition. Simulations demonstrate the efficiency and correctness of our algorithm.     

Routing-efficient CDS construction in Disk-Containment Graphs

In wireless networks, Connected Dominating Sets (CDSs) are widely used as virtual backbones for communications. On one hand, reducing the backbone size will reduce the maintenance overhead. So how to minimize the CDS size is a critical issue. On the other hand, when evaluating the performance of a wireless network, the hop distance between two communication nodes, which reflect the energy consumption and response delay, is of great importance. Hence how to minimize the routing cost is also a key problem for constructing the network virtual backbone. In this paper, we study the problem of constructing applicable CDS in wireless networks in terms of size and routing cost. We formulate a wireless network as a Disk-Containment Graph (DCG), which is a generalization of the Unit-Disk Graph (UDG), and we develop an efficient algorithm to construct CDS in such kind of graphs. The algorithm contains two parts and is flexible to balance the performance between the two metrics. We also analyze the algorithm theoretically. It is shown that our algorithm has provable performance in minimizing the CDS size and reducing the communication distance for routing.     

Modeling and optimization of energy efficient routing in wireless sensor networks

Wireless sensor networks (WSNs) have important applications in remote environmental monitoring and target tracking. The development of WSNs in recent years has been facilitated by the availability of sensors that are smaller, less expensive, and more intelligent. The design of a WSN depends significantly on its desired applications and must take into account factors such as the environment, the design objectives of the application, the associated costs, the necessary hardware, and any applicable system constraints. In this study, we propose mathematical models for a routing protocol (network design) under particular resource restrictions within a wireless sensor network. We consider two types of constraints: the distance between the linking sensors and the energy used by the sensors. The proposed models aim to identify energy-efficient paths that minimize the energy consumption of the network from the source sensor to the base station. The computational results show that the presented models can be used efficiently and applied to other network design contexts with resource restrictions (e.g., to multi-level supply chain networks).     

Topology design for on-demand dual-path routing in wireless networks

One way to achieve reliability with low-latency is through multi-path routing and transport protocols that build redundant delivery channels (or data paths) to reduce end-to-end packet losses and retransmissions. However, the applicability and effectiveness of such protocols are limited by the topological constraints of the underlying communication infrastructure. Multiple data delivery paths can only be constructed over networks that are capable of supporting multiple paths. In mission-critical wireless networks, the underlying network topology is directly affected by the terrain, location and environmental interferences, however the settings of the wireless radios at each node can be properly configured to compensate for these effects for multi-path support. In this work we investigate optimization models for topology designs that enable end-to-end dual-path support on a distributed wireless sensor network. We consider the case of a fixed sensor network with isotropic antennas, where the control variable for topology management is the transmission power on network nodes. For optimization modeling, the network metrics of relevance are coverage, robustness and power utilization. The optimization models proposed in this work eliminate some of the typical assumptions made in the pertinent network design literature that are too strong in this application context.     

多无线WMN中干扰最小化信道分配算法研究

为了提高无线迈适网的通信容量,网络中的每个路由节点均配备有多个无线网卡,并提供多个可用的无线信道.如何将这些信道合理地分配到网络的各个通信链路上,使得整个网络的干扰最小是一个至关重要问题.分析了基于禁忌搜索的信道分配算法,并针对该算法存在的问题,提出了初步的改进算法.     

Virtual path layouts optimizing total hop count on ATM tree networks

This paper presents some algorithmic results concerning virtual path layouts for the one-to-many communication problem in ATM tree networks. The ATM network model is based on covering the network with a layout of virtual paths, under some constraints on the allowed load, namely, the number of paths that can share an edge. The quality measure used is the hop count, namely, the number of edges traversed between two vertices that need to communicate. Whereas most former results concerned the maximum hop count of the virtual path layout, our interest here is in measuring its total hop count, or alternatively its average hop count. The paper presents a dynamic programming algorithm for planning ATM network layouts with minimal total hop count for one-to-many requirements under load constraints over the class of tree networks.     

马氏模型下移动自组网随选型路由协议特性分析

移动自组网络（简称MANET）因其移动性及无基础设施支持等特点已经成为无线通信网络中的热门问题．通过将一个MANET网络中每条链边的长度看作一个生灭过程,并且假设在泛洪过程中空间可以复用n次,建立了移动自组网络空间可复用的马氏模型,简记为n-SRBDM．在一个典型的随选型路由协议即动态源路由（DSR）协议的基础上,研究了网络的一些关键性能参数,给出了路由泛洪距离的概率分布和期望,限定泛洪步数时成功寻路的概率、发现τ-时有效路径及对称有效路径的概率,发现一条有效路径的平均时间等,对于路由维护过程,也引入并研究了一些网络性能参数,例如,路由恢复的平均频率,路由有效的平均时间．对于这些网络参数在空间可复用和空间不可复用两种情形下进行了比较．证明了空间可复用模型下的路由选择更为有效．     

Optimal infrastructure design and expansion of broadband wireless access networks

This paper investigates the important infrastructure design and expansion problem for broadband wireless access networks subject to user demand constraints and system capacity constraints. For the problem, an integer program is derived and a heuristic solution procedure is proposed based on Lagrangean relaxation. In the computational experiments, our Lagrangean relaxation based algorithm can solve this complex design and expansion problem quickly and near optimally. Based on the test results, it is suggested that the proposed algorithm may be practically used for the infrastructure design and expansion problem for broadband wireless access networks.     

Polyhedral results for the edge capacity polytope

Network loading problems occur in the design of telecommunication networks, in many different settings. For instance, bifurcated or non-bifurcated routing (also called splittable and unsplittable) can be considered. In most settings, the same polyhedral structures return. A better understanding of these structures therefore can have a major impact on the tractability of polyhedral-guided solution methods. In this paper, we investigate the polytopes of the problem restricted to one arc/edge of the network (the undirected/directed edge capacity problem) for the non-bifurcated routing case.?As an example, one of the basic variants of network loading is described, including an integer linear programming formulation. As the edge capacity problems are relaxations of this network loading problem, their polytopes are intimately related. We give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope. We describe classes of strong valid inequalities for the edge capacity polytopes, and we derive conditions under which these constraints define facets. For the diverse classes the complexity of lifting projected variables is stated.?The derived inequalities are tested on (i) the edge capacity problem itself and (ii) the described variant of the network loading problem. The results show that the inequalities substantially reduce the number of nodes needed in a branch-and-cut approach. Moreover, they show the importance of the edge subproblem for solving network loading problems. Received: September 2000 / Accepted: October 2001?Published online March 27, 2002     

On routing in VLSI design and communication networks

In this paper, we study the global routing problem in VLSI design and the multicast routing problem in communication networks. First we propose new and realistic models for both problems. In the global routing problem in VLSI design, we are given a lattice graph and subsets of the vertex set. The goal is to generate trees spanning these vertices in the subsets to minimize a linear combination of overall wirelength (edge length) and the number of bends of trees with respect to edge capacity constraints. In the multicast routing problem in communication networks, a graph is given to represent the network, together with subsets of the vertex set. We are required to find trees to span the given subsets and the overall edge length is minimized with respect to capacity constraints. Both problems are APX-hard. We present the integer linear programming (LP) formulation of both problems and solve the LP relaxations by the fast approximation algorithms for min-max resource-sharing problems in [K. Jansen, H. Zhang, Approximation algorithms for general packing problems and their application to the multicast congestion problem, Math. Programming, to appear, doi:10.1007/s10107-007-0106-8] (which is a generalization of the approximation algorithm proposed by Grigoriadis and Khachiyan [Coordination complexity of parallel price-directive decomposition, Math. Oper. Res. 2 (1996) 321-340]). For the global routing problem, we investigate the particular property of lattice graphs and propose a combinatorial technique to overcome the hardness due to the bend-dependent vertex cost. Finally, we develop asymptotic approximation algorithms for both problems with ratios depending on the best known approximation ratio for the minimum Steiner tree problem. They are the first known theoretical approximation bound results for the problems of minimizing the total costs (including both the edge and the bend costs) while spanning all given subsets of vertices.     

Robust delay-constrained routing in telecommunications

In telecommunications, operators usually use market surveys and statistical models to estimate traffic evolution in networks or to approximate queuing delay functions in routing strategies. Many research activities concentrated on handling traffic uncertainty in network design. Measurements on real world networks have shown significant errors in delay approximations, leading to weak management decisions in network planning. In this work, we introduce elements of robust optimization theory for delay modeling in routing problems. Different types of data uncertainty are considered and linked to corresponding robust models. We study a special case of constraints featuring separable additive functions. Specifically, we consider that each term of the sum is disturbed by a random parameter. These constraints are frequent in network based problems, where functions reflecting real world measurements on links are summed up over end-to-end paths. While classical robust formulations have to deal with the introduction of new variables, we show that, under specific hypotheses, the deterministic robust counterpart can be formulated in the space of original variables. This offers the possibility of constructing tractable robust models. Starting from Soyster’s conservative model, we write and compare different uncertainty sets and formulations offering each a different protection level for the delay constrained routing problem. Computational experiments are developed in order to evaluate the “price of robustness” and to assess the quality of the new formulations.     

Maximizing lifetime of a wireless sensor network via joint optimizing sink placement and sensor-to-sink routing

Wireless sensor networks typically contain hundreds of sensors. The sensors collect data and relay it to sinks through single hop or multiple hop paths. Sink deployment significantly influences the performance of a network. Since the energy capacity of each sensor is limited, optimizing sink deployment and sensor-to-sink routing is crucial. In this paper, this problem is modeled as a mixed integer optimization problem. Then, a novel layer-based diffusion particle swarm optimization method is proposed to solve this large-scaled optimization problem. In particular, two sensor-to-sink binding algorithms are combined as inner layer optimization to evaluate the fitness values of the solutions. Compared to existing methods that the sinks are selected from candidate positions, our method can achieve better performance since they can be placed freely within a geometrical plane. Several numerical examples are used to validate and demonstrate the performance of our method. The reported numerical results show that our method is superior to those existing. Furthermore, our method has good scalability which can be used to deploy a large-scaled sensor network.     

Parameterized complexity of spare capacity allocation and the multicost Steiner subgraph problem

We study the computational complexity of the Spare Capacity Allocation problem arising in optical networks that use a shared mesh restoration scheme. In this problem we are given a network with edge capacities and point-to-point demands, and the goal is to allocate two edge-disjoint paths for each demand (a working path and a so-called restoration path, which is activated only if the working path fails) so that the capacity constraints are satisfied and the total cost of the used and reserved bandwidth is minimized. We focus on the setting where we deal with a group of demands together, and select their restoration paths simultaneously in order to minimize the total cost. We investigate how the computational complexity of this problem is affected by certain parameters, such as the number of restoration paths to be selected, or the treewidth of the network graph. To analyze the complexity of the problem, we introduce a generalization of the Steiner Forest problem that we call Multicost Steiner Subgraph. We study its parameterized complexity, and identify computationally easy and hard cases by providing hardness proofs as well as efficient (fixed-parameter tractable) algorithms.     

Hop Constrained Network Design Problem with Partial Survivability

This paper studies the hop constrained network design problem with partial survivability, namely, given an undirected network, a set of point-to-point demands (commodities), and transmission link costs, identify two node disjoint paths for each demand (commodity) to minimize the total costs subject to the constraints that each demand is routed and traverses at most a specified number of links (or hops) on both the paths.A mathematical programming formulation of the problem is presented and an efficient solution procedure based on the linear programming relaxation is developed. Extensive computational results across a number of networks are reported. These results indicate that the solution procedure is effective for a wide range of problem sizes.     

Better Optimization of Nonlinear Uncertain Systems (BONUS): A New Algorithm for Stochastic Programming Using Reweighting through Kernel Density Estimation

Given a series-parallel queueing network topology with exponential servers of finite capacity, a systematic design methodology is presented that approximately solves the optimal routing and buffer space allocation problems within the network. The multi-objective stochastic nonlinear programming problem in integer variables is described and a two-stage iterative optimization procedure is presented which interconnects the routing and buffer space allocation problems. The algorithmic procedure couples the Expansion method, a decomposition method for computing performance measures in queueing networks with finite capacity, along with Powell's unconstrained optimization procedure which allocates the buffers and a multi-variable search procedure for determining the routing probabilities. The effectiveness and efficiency of the resulting two-stage design methodology is tested and evaluated in a series of experimental designs along with simulations of the network topologies.     

Optimal multicast route packing

This paper considers the hop-constrained multicast route packing problem with a bandwidth reservation to build QoS-guaranteed multicast routing trees with a minimum installation cost. Given a set of multicast sessions, each of which has a hop limit constraint and a bandwidth requirement, the problem is to determine the set of multicast routing trees in an arc-capacitated network with the objective of minimizing the cost. For the problem, we propose a branch-and-cut-and-price algorithm, which can be viewed as a branch-and-bound method incorporating both the strong cutting plane algorithm and the column generation method. We implemented and tested the proposed algorithm on randomly generated problem instances with sizes up to 30 nodes, 570 arcs, and 10 multicast sessions. The test results show that the algorithm can obtain the optimal solution to practically sized problem instances within a reasonable time limit in most cases.     

Two-level evolutionary approach to the survivable mesh-based transport network topological design

The complete topology design problem of survivable mesh-based transport networks is to address simultaneously design of network topology, working path routing, and spare capacity allocation based on span-restoration. Each constituent problem in the complete design problem could be formulated as an Integer Programming (IP) and is proved to be NP\mathcal{NP} -hard. Due to a large amount of decision variables and constraints involved in the IP formulation, to solve the problem directly by exact algorithms (e.g. branch-and-bound) would be impractical if not impossible. In this paper, we present a two-level evolutionary approach to address the complete topology design problem. In the low-level, two parameterized greedy heuristics are developed to jointly construct feasible solutions (i.e., closed graph topologies satisfying all the mesh-based network survivable constraints) of the complete problem. Unlike existing “zoom-in”-based heuristics in which subsets of the constraints are considered, the proposed heuristics take all constraints into account. An estimation of distribution algorithm works on the top of the heuristics to tune the control parameters. As a result, optimal solution to the considered problem is more likely to be constructed from the heuristics with the optimal control parameters. The proposed algorithm is evaluated experimentally in comparison with the latest heuristics based on the IP software CPLEX, and the “zoom-in”-based approach on 28 test networks problems. The experimental results demonstrate that the proposed algorithm is more effective in finding high-quality topologies than the IP-based heuristic algorithm in 21 out of 28 test instances with much less computational costs, and performs significantly better than the “zoom-in”-based approach in 19 instances with the same computational costs.     

Processor interconnection networks from Cayley graphs

Cayley graphs of groups are presently being considered by the computer science community as models of architectures for large scale parallel processor computers. In the first section of this paper we discuss Cayley graphs and show how they may be used as a tool for the design and analysis of network architectures for these types of computers.

Observing that routing on a Cayley graph is equivalent to a certain factoring problem in the associated group, we have been able to use a known powerful factoring technique in computational group theory to produce a fast efficient routing algorithm on the associated Cayley graph. In the second section of this paper we present this work. This research can be regarded as a first attempt to find general purpose routing algorithms for interconnection networks.

Believing that average diameter of a network for a large scale MIMD machine is the predominant factor in determining network performance, we designed Cayley graphs to be used in a special study performed at the Supercomputing Research Center (SRC). The importance of the average diameter in determining network performance was supported by the fact that the graphs found by us had the smallest average diameter and outperformed all other graphs evaluated in the study. In fact, before being driven into saturation, one of our graphs sustained 9.4% more network traffic than the next best candidate, a butterfly architecture, and 74.3% better than the bench mark 2-d mesh. The last section of our paper is devoted to this work.

This paper is divided into three sections. In the first section we discuss Cayley graphs and show how they may be used as a tool for the design and analysis of network architectures for parallel computers. In the second section we present our research on the routing problem. This research can be regarded as a first attempt to find general purpose routing algorithms for interconnection networks. In the last section we present some evidence that average diameter of a network for a large scale MIMD machine is the predominant factor in determining network performance.     

Multicommodity flow models for spanning trees with hop constraints

In this paper we compare the linear programming relaxations of undirected and directed multicommodity flow formulations for the terminal layout problem with hop constraints. Hop constraints limit the number of hops (links) between the computer center and any terminal in the network. These constraints model delay constraints since a smaller number of hops decreases the maximum delay transmission time in the network. They also model reliability constraints because with a smaller number of hops there is a lower route loss probability. Hop constraints are easily modelled with the variables involved in multicommodity flow formulations. We give some empirical evidence showing that the linear programming relaxation of such formulations give sharp lower bounds for this hop constrained network design problem. On the other hand, these formulations lead to very large linear programming models. Therefore, for bounding purposes we also derive several lagrangean based procedures from a directed multicommodity flow formulation and present some computational results taken from a set of instances with up to 40 nodes.