On Network Coding Capacity - Matroidal Networks and Network Capacity Regions

One fundamental problem in the field of network coding is to determine the network coding capacity of networks under various network coding schemes. In this thesis, we address the problem with two approaches: matroidal networks and capacity regions. In our matroidal approach, we prove the converse of the theorem which states that, if a network is scalar-linearly solvable then it is a matroidal network associated with a representable matroid over a finite field. As a consequence, we obtain a correspondence between scalar-linearly solvable networks and representable matroids over finite fields in the framework of matroidal networks. We prove a theorem about the scalar-linear solvability of networks and field characteristics. We provide a method for generating scalar-linearly solvable networks that are potentially different from the networks that we already know are scalar-linearly solvable. In our capacity region approach, we define a multi-dimensional object, called the network capacity region, associated with networks that is analogous to the rate regions in information theory. For the network routing capacity region, we show that the region is a computable rational polytope and provide exact algorithms and approximation heuristics for computing the region. For the network linear coding capacity region, we construct a computable rational polytope, with respect to a given finite field, that inner bounds the linear coding capacity region and provide exact algorithms and approximation heuristics for computing the polytope. The exact algorithms and approximation heuristics we present are not polynomial time schemes and may depend on the output size.Comment: Master of Engineering Thesis, MIT, September 2010, 70 pages, 10 figure

Computing Bounds on Network Capacity Regions as a Polytope Reconstruction Problem

We define a notion of network capacity region of networks that generalizes the notion of network capacity defined by Cannons et al. and prove its notable properties such as closedness, boundedness and convexity when the finite field is fixed. We show that the network routing capacity region is a computable rational polytope and provide exact algorithms and approximation heuristics for computing the region. We define the semi-network linear coding capacity region, with respect to a fixed finite field, that inner bounds the corresponding network linear coding capacity region, show that it is a computable rational polytope, and provide exact algorithms and approximation heuristics. We show connections between computing these regions and a polytope reconstruction problem and some combinatorial optimization problems, such as the minimum cost directed Steiner tree problem. We provide an example to illustrate our results. The algorithms are not necessarily polynomial-time.Comment: Appeared in the 2011 IEEE International Symposium on Information Theory, 5 pages, 1 figur

Optimization in Telecommunication Networks

Network design and network synthesis have been the classical optimization problems intelecommunication for a long time. In the recent past, there have been many technologicaldevelopments such as digitization of information, optical networks, internet, and wirelessnetworks. These developments have led to a series of new optimization problems. Thismanuscript gives an overview of the developments in solving both classical and moderntelecom optimization problems.We start with a short historical overview of the technological developments. Then,the classical (still actual) network design and synthesis problems are described with anemphasis on the latest developments on modelling and solving them. Classical results suchas Menger’s disjoint paths theorem, and Ford-Fulkerson’s max-flow-min-cut theorem, butalso Gomory-Hu trees and the Okamura-Seymour cut-condition, will be related to themodels described. Finally, we describe recent optimization problems such as routing andwavelength assignment, and grooming in optical networks.operations research and management science;

Path Planning for Cooperative Routing of Air-Ground Vehicles

We consider a cooperative vehicle routing problem for surveillance and reconnaissance missions with communication constraints between the vehicles. We propose a framework which involves a ground vehicle and an aerial vehicle; the vehicles travel cooperatively satisfying the communication limits, and visit a set of targets. We present a mixed integer linear programming (MILP) formulation and develop a branch-and-cut algorithm to solve the path planning problem for the ground and air vehicles. The effectiveness of the proposed approach is corroborated through extensive computational experiments on several randomly generated instances

A large neighbourhood based heuristic for two-echelon routing problems

In this paper, we address two optimisation problems arising in the context of city logistics and two-level transportation systems. The two-echelon vehicle routing problem and the two-echelon location routing problem seek to produce vehicle itineraries to deliver goods to customers, with transits through intermediate facilities. To efficiently solve these problems, we propose a hybrid metaheuristic which combines enumerative local searches with destroy-and-repair principles, as well as some tailored operators to optimise the selections of intermediate facilities. We conduct extensive computational experiments to investigate the contribution of these operators to the search performance, and measure the performance of the method on both problem classes. The proposed algorithm finds the current best known solutions, or better ones, for 95% of the two-echelon vehicle routing problem benchmark instances. Overall, for both problems, it achieves high-quality solutions within short computing times. Finally, for future reference, we resolve inconsistencies between different versions of benchmark instances, document their differences, and provide them all online in a unified format

Kiertovaihtoalgoritmi ja muunnoksia yleistetylle ajoneuvoreititysongelmalle

Vehicle routing problems have numerous applications in fields such as transportation, supply logistics and network design. The optimal design of these routes fall in the category of NP-hard optimization problems, meaning that the computational complexity increases extremely fast with increasing problem size. The Generalized Vehicle Routing Problem (GVRP) is a general problem type that includes a broad variety of other problems as special cases. The main special feature of the GVRP is that the customers are grouped in clusters. For each cluster, only one customer is visited. In this thesis, we implement a heuristic algorithm to solve GVRP instances in reasonable time. Especially, we include a cyclic exchange method that considers a very large search neighborhood. In addition, we study the related Capacitated m-Ring-Star Problem (CmRSP). We present the Distance-Constrained Capacitated m-Ring-Star Problem (DCmRSP) and show that it contains the Multivehicle Covering Tour Problem (MCTP) as a special case. We show that DCmRSP instances can be transformed to (distance-constrained) GVRP with minor adaptations and solved with the same heuristic algorithm. Our algorithm is able to find best known solutions to all GVRP test instances; for two of them, our method shows strict improvement. The transformed CmRSP and MCTP instances are solved successfully by the same algorithm with adequate performance.Ajoneuvoreititysongelmilla on lukuisia sovelluksia muun muassa logistiikan ja verkostosuunnittelun aloilla. Tällaisten reittien optimaalinen ratkaiseminen kuuluu NP-vaikeiden optimointiongelmien kategoriaan, eli ratkaisuun vaadittava laskentateho kasvaa erittäin nopeasti ongelman koon suhteen. Yleistetty ajoneuvoreititysongelma (Generalized Vehicle Routing Problem, GVRP) on ongelmatyyppi, joka kattaa joukon muita reititysongelmia erikoistapauksina. GVRP:n selkein erityispiirre on asiakkaiden jako ryppäisiin: kussakin ryppäässä on käytävä tasan yhden asiakkaan luona. Tässä diplomityössä esitellään ja toteutetaan heuristinen algoritmi, joka etsii kohtalaisessa ajassa ratkaisuja GVRP-ongelmiin. Menetelmä sisältää kiertovaihtoalgoritmin, joka kykenee etsimään ratkaisuja hyvin laajasta ympäristöstä. Tutkimuksen kohteena on lisäksi m-rengastähtiongelma (Capacitated m-Ring-Star Problem, CmRSP). Esittelemme ongelman etäisyysrajoitetun version (DCmRSP), ja näytämme, että kyseiseen ongelmaan sisältyy usean ajoneuvon peittävän reitin ongelma (Multivehicle Covering Tour Problem). Näytämme, että DCmRSP-ongelman pystyy pienin muutoksin muuntamaan GVRP-ongelmaksi ja ratkaisemaan samalla heuristisella algoritmilla. Algoritmi löytää parhaat tunnetut ratkaisut kaikkiin GVRP-testitehtäviin. Kahdessa tapauksessa ratkaisu on parempi aiemmin löydettyihin nähden. Algoritmi kykenee ratkaisemaan muunnetut CmRSP- ja MCTP-testitehtävät kohtalaisella ratkaisulaadulla

Topological Design of Survivable Networks

In the field of telecommunications there are several ways of establishing links between different physical places that must be connected according to the characteristics and the type of service they should provide. Two main considerations to be taken into account and which require the attention of the network planners are, in one hand the economic effort necessary to build the network, and in the other hand the resilience of the network to remain operative in the event of failure of any of its components. A third consideration, which is very important when quality of services required, such as video streaming or communications between real-time systems, is the diameter constrained reliability. In this thesis we study a set of problems that involve such considerations. Firstly, we model a new combinatorial optimization problem called Capacitated m-Two Node Survivable Star Problem (CmTNSSP). In such problem we optimize the costs of constructing a network composed of 2-node-connected components that converge in a central node and whose terminals can belong to these connected 2-node structures or be connected to them by simple edges. The CmTNSSP is a relaxation of the Capacitated Ring Star Problem (CmRSP), where the cycles of the latter can be replaced by arbitrary 2-node-connected graphs. According to previous studies, some of the structural properties of 2-node-connected graphs can be used to show a potential improvement in construction costs, over solutions that exclusively use cycles. Considering that the CmTNSSP belongs to the class of NP-Hard computational problems, a GRASP-VND metaheuristic was proposed and implemented for its approximate resolution, and a comparison of results was made between both problems (CmRSP and CmTNSSP) for a series of instances. Some local searches are based on exact Integer Linear Programming formulations. The results obtained show that the proposed metaheuristic reaches satisfactory levels of accuracy, attaining the global optimum in several instances. Next, we introduce the Capacitated m Ring Star Problem under Diameter Constrained Reliability (CmRSP-DCR) wherein DCR is considered as an additional restriction, limiting the number of hops between nodes of the CmRSP problem and establishing a minimum level of network reliability. This is especially useful in networks that should guarantee minimum delays and quality of service. The solutions found in this problem can be improved by applying some of the results obtained in the study of the CmTNSSP. Finally, we introduce a variant of the CmTNSSP named Capacitated Two-Node Survivable Tree Problem, motivated by another combinatorial optimization problem most recently treated in the literature, called Capacitated Ring Tree Problem (CRTP). In the CRTP, an additional restriction is added with respect to CmRSP, where the terminal nodes are of two different types and tree structures are also allowed. Each node in the CRTP may be connected exclusively in one cycle, or may be part of a cycle or a tree indistinctly, depending on the type of node. In the variant we introduced, the cycles are replaced by 2-node-connected structures. This study proposes and implements a GRASP-VND metaheuristic with specific local searches for this type of structures and adapts some of the exact local searches used in the resolution CmTNSSP. A comparison of the results between the optimal solutions obtained for the CRTP and the CTNSTP is made. The results achieved show the robustness and efficiency of the metaheuristi

The capacitated m two node survivable star problem

The problem addressed in this paper attempts to efficiently solve a network design with redundant connections, often used by telephone operators and internet services. This network connects customers with one master node and sets some rules that shape its construction, such as number of customers, number of components and types of links, in order to meet operational needs and technical constraints. We propose a combinatorial optimization problem called CmTNSSP (Capacitated m Two- Node-Survivable Star Problem), a relaxation of CmRSP (Capacitated m Ring Star Problem). In this variant of CmRSP the rings are not constrained to be cycles; instead, they can be two node connected components. The contributions of this paper are (a) introduction and definition of a new problem (b) the specification of a mathematical programming model of the problem to be treated, and (c) the approximate resolution thereof through a GRASP metaheuristic, which alternates local searches that obtain incrementally better solutions, and exact resolution local searches based on mathematical programming models, particularly Integer Linear Programming ones. Computational results obtained by developed algorithms show robustness and competitiveness when compared to results of the literature relative to benchmark instances. Likewise, the experiments show the relevance of considering the specific variant of the problem studied in this work

Heuristic Solution to Protect Communications in WDM Networks using P-cycles

Optical WDM mesh networks are able to transport huge amount of information. The use of such technology however poses the problem of protection against failures such as fibre cuts. One of the principal methods for link protection used in optical WDM networks is pre-configured protection cycle (p-cycle). The major problem of this method of protection resides in finding the optimal set of p-cycles which protect the network for a given distribution of working capacity. Existing heuristics generate a large set of p-cycle candidates which are entirely independent of the network state, and from then the good sub-set of p-cycles which will protect the network is selected. In this paper, we propose a new algorithm of generation of p-cycles based on the incremental aggregation of the shortest cycles. Our generation of p-cycles depends on the state of the network. This enables us to choose an efficient set of p-cycles which will protect the network. The set of p-cycles that we generate is the final set which will protect the network, in other words our heuristic does not go through the additional step of p-cycle selectio

A Column Generation Based Heuristic for the Multicommodity-ring Vehicle Routing Problem

AbstractWe study a new routing problem arising in City Logistics. Given a ring connecting a set of urban distribution centers (UDCs) in the outskirts of a city, the problem consists in delivering goods from virtual gates located outside the city to the customers inside of it. Goods are transported from a gate to a UDC, then either go to another UDC before being delivered to customers or are directly shipped from the first UDC. The reverse process occurs for pick-up. Routes are performed by electric vans and may be open. The objective is to find a set of routes that visit each customer and to determine ring and gates-UDC flows so that the total transportation and routing cost is minimized. We solve this problem using a column generation-based heuristic, which is tested over a set of benchmark instances issued from a more strategic location-routing problem