Consider the following classical network design problem: a set of terminals T: {t.i} wants to send traffic to a "root" r in an 'x-node graph G: (V, E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocated on the edges to permit this. However, bandwidth on an edge e can only be allocated in integral multiples of some base capacity ue-- and hence provisioning k x ue bandwidth on edge e incurs a cost of [k] times the cost of that edge. The objective is a minimum-cost feasible solution. This is one of many network design problems widely studied where the bandwidth allocation being governed by side constraints: edges may only allow a subset of cables to be purchased on them, or certain quality-of-service requirements may have to be met. In this work, we show that the above problem, and in fact, several basic problems in this general network design framework, cannot be approximated better than ~(log log n) unless NP c _ OTIME(,r~°(l°gl°gl°gn)). In particular, we show that this inapproximability threshold holds for (i) the Priority-Steiner Tree problem [7], (ii) the Cost-Distance problem [31], and the single-sink version of an even more fundamental problem, (iii) Fixed Charge Network Flow [33]. Our results provide a further breakthrough in the understanding of the level of complexity of network design problems. These are the first non-constant hardness results known for all these problems.

This book was typeset with TEX using L ATEX and many further formatting packages. The pictures were prepared using pstricks, xfig, gnuplot and gmt. All numerals in this text are recycled. Für meine Eltern Preface Avoid reality at all costs — fortune(6) As the inclined reader will find out soon enough, this thesis is not about deeply involved mathematics as a mean in itself, but about how to apply mathematics to solve real-world problems. We will show how to shape, forge, and yield our tool of choice to rapidly answer questions of concern to people outside the world of mathematics. But there is more to it. Our tool of choice is software. This is not unusual, since it has become standard practice in science to use software as part of experiments and sometimes even for proofs. But in order to call an experiment scientific it must be reproducible. Is this the case?

Abstract—We estimate potential energy savings in IP-over-WDM networks achieved by switching off router line cards in low-demand hours. We compare three approaches to react on dynamics in the IP traffic over time, FUFL, DUFL and DUDL. They provide different levels of freedom in adjusting the routing of lightpaths in the WDM layer and the routing of demands in the IP layer. Using MILP models based on realistic network topologies and node architectures as well as realistic demands, power, and cost values, we show that already a simple monitoring of the lightpath utilization in order to deactivate empty line cards (FUFL) brings substantial benefits. The most significant savings, however, are achieved by rerouting traffic in the IP layer (DUFL), which allows emptying and deactivating lightpaths together with the corresponding line cards. A sophisticated reoptimization of the virtual topologies and the routing in the optical domain for every demand scenario (DUDL)yields nearly no additional profits in the considered networks. I.

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Consider the following classical network design problem: a set of terminals T = {t_i} wants to send traffic to a “root” in an-node graph����. Each terminal�sends ��units of traffic, and enough bandwidth has to be allocated on the edges to permit this. However, bandwidth on an edge �can only be allocated in integral multiples of some base capacity� — and hence provisioning � �bandwidth on edge�incurs a cost of���times the cost of that edge. The objective is a minimum-cost feasible solution. This is one of many network design problems widely studied where the bandwidth allocation being governed by side constraints: edges may only allow a subset of cables to be purchased on them, or certain quality-of-service requirements may have to be met. In this work, we show that the above problem, and in fact, several basic problems in this general network design framework, cannot be approximated better thanÅ�� unless NP�DTIME ���. In particular, we show that this inapproximability threshold holds for (i) the Priority-Steiner Tree problem [7], (ii) the Cost-Distance problem [31], and the single-sink version of an even more fundamental problem, (iii) Fixed Charge Network Flow [33]. Our results provide a further breakthrough in the understanding of the level of complexity of network design problems. These are the first non-constant hardness results known for all these problems.

Abstract — This paper presents research on the Vehicle Routing Problem (VRP) using a sweep heuristic method with 2-opt exchange and traveling salesman tours and an integer programming model for split delivery VRP model to select the best route to pick up and delivery customers from/to desired destination and depot. The modeling language, AMPL with CPLEX is used to develop the model and implement the sweep heuristic. The research found that the integer programming model produced the optimal result for some cases and failed to produce the optimal result for some cases. However, the sweep heuristic gave good solutions for all cases within a small amount of computational time. The research also investigated sensitivity analysis with respect to the vehicle capacity. The results indicate a savings in number of vehicle used with a small increase in vehicle capacity. The case study of University of The Thai Chamber of Commerce (UTCC) which provides bus services to pick up and deliver staff from/to home and university is selected to present in this paper.

Consider the following classical network design problem: a set of terminals T: {t.i} wants to send traffic to a "root" r in an 'x-node graph G: (V, E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocated on the edges to permit this. However, bandwidth on an edge e can only be allocated in integral multiples of some base capacity ue-- and hence provisioning k x ue bandwidth on edge e incurs a cost of [k] times the cost of that edge. The objective is a minimum-cost feasible solution. This is one of many network design problems widely studied where the bandwidth allocation being governed by side constraints: edges may only allow a subset of cables to be purchased on them, or certain quality-of-service requirements may have to be met. In this work, we show that the above problem, and in fact, several basic problems in this general network design framework, cannot be approximated better than ~(log log n) unless NP c _ OTIME(,r~°(l°gl°gl°gn)). In particular, we show that this inapproximability threshold holds for (i) the Priority-Steiner Tree problem [7], (ii) the Cost-Distance problem [31], and the single-sink version of an even more fundamental problem, (iii) Fixed Charge Network Flow [33]. Our results provide a further breakthrough in the understanding of the level of complexity of network design problems. These are the first non-constant hardness results known for all these problems. 1

This thesis presents approximation algorithms for some N P-Hard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widely-believed complexity-theoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomial-time) algorithms that solve these problems exactly. Hence, if one desires efficient algorithms for such problems, it is necessary to consider approximate solutions: An approximation algorithm for an N P-Hard problem is a polynomial time algorithm which, for any instance of the problem, finds a solution whose value is guaranteed to be within a multiplicative factor ρ of the value of an optimal solution to that instance. We attempt to design algorithms for which this factor ρ, referred to as the approximation ratio of the algorithm, is as small as possible. The field of Network Design comprises a large class of problems that deal with constructing networks of low cost and/or high capacity, routing data through existing networks, and many related issues. In this thesis, we focus chiefly on designing fault-tolerant networks. Two vertices u, v in a network are said to be k-edge-connected if deleting any set of k − 1 edges leaves u and v connected; similarly, they are k-vertex connected if deleting any set of k − 1 other vertices or edges leaves u and v connected. We focus on building networks that are highly connected, meaning