Optimal Trees

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Solving two-stage stochastic network design problems to optimality

The Steiner tree problem (STP) is a central and well-studied graph-theoretical combinatorial optimization problem which plays an important role in various applications. It can be stated as follows: Given a weighted graph and a set of terminal vertices, find a subset of edges which connects the terminals at minimum cost. However, in real-world applications the input data might not be given with certainty or it might depend on future decisions. For the STP, for example, edge costs representing the costs of establishing links may be subject to inflations and price deviations. In this thesis we tackle data uncertainty by using the concept of stochastic programming and we study the two-stage stochastic version of the Steiner tree problem (SSTP). Thereby, a set of scenarios defines the possible outcomes of a random variable; each scenario is given by its realization probability and defines a set of terminals and edge costs. A feasible solution consists of a subset of edges in the first stage and edge subsets for all scenarios (second stage) such that each terminal set is connected. The objective is to find a solution that minimizes the expected cost. We consider two approaches for solving the SSTP to optimality: combinatorial algorithms, in particular fixed-parameter tractable (FPT) algorithms, and methods from mathematical programming. Regarding the combinatorial algorithms we develop a linear-time algorithm for trees, an FPT algorithm parameterized by the number of terminals, and we consider treewidth-bounded graphs where we give the first FPT algorithm parameterized by the combination of treewidth and number of scenarios. The second approach is based on deriving strong integer programming (IP) formulations for the SSTP. By using orientation properties we introduce new semi-directed cut- and flow-based IP formulations which are shown to be stronger than the undirected models from the literature. To solve these models to optimality we use a decomposition-based two-stage branch&cut algorithm, which is improved by a fast and efficient method for strengthening the optimality cuts. Moreover, we develop new and stronger integer optimality cuts. The computational performance is evaluated in a comprehensive computational study, which shows the superiority of the new formulations, the benefit of the decomposition, and the advantage of using the strengthened optimality cuts. The Steiner forest problem (SFP) is a related problem where sets of terminals need to be connected. On the one hand, the SFP is a generalization of the STP and on the other hand, we show that the SFP is a special case of the SSTP. Therefore, our results are transferable to the SFP and we present the first FPT algorithm for treewidth-bounded graphs and we model new and stronger (semi-)directed cut- and flow-based IP formulations for the SFP. In the second part of this thesis we consider the two-stage stochastic survivable network design problem, an extension of the SSTP where pairs of vertices may demand a higher connectivity. Similarly to the first part we introduce new and stronger semi-directed cut-based models, apply the same decomposition along with the cut strengthening technique, and argue the validity of the newly introduced integer optimality cuts. A computational study shows the benefit, robustness, and good performance of the decomposition and the cut strengthening method

Designing data center networks for high throughput

Data centers with tens of thousands of servers now support popular Internet services, scientific research, as well as industrial applications. The network is the foundation of such facilities, giving the large server pool the ability to work together on these applications. The network needs to provide high throughput between servers to ensure that computations are not slowed down by network bottlenecks, with servers waiting on data from other servers. This work address two broad, related questions about high-throughput data center network design: (a) how do we measure and benchmark various network designs for throughput? and (b) how do we design such networks for near-optimal throughput? The problem of designing high-throughput networks has received a lot of attention, with multiple interesting architectures being proposed every year. However, there is no clarity on how one should benchmark these networks and how they compare to each other. In fact, this work shows that commonly used measurement approaches, in particular, cut-metrics like bisection bandwidth, do not predict throughput accurately. In contrast, we directly evaluate the throughput of networks on both uniform and (heretofore unknown) nearly-worst-case traffic matrices, and include here a comparison of 10 networks using this approach. Further, prior work has not addressed a fundamental question: how far are we from throughput-optimal design? In this work, we propose the first upper bound on network throughput for any topology with identical switches. Although designing optimal topologies is infeasible, we demonstrate that random graphs achieve throughput surprisingly close to this bound -- within a few percent at the scale of a few thousand servers for uniform traffic. Our approach also addresses important practical concerns in the design of data center networks, such as incremental expansion and heterogeneous design – as more and varied equipment is added to a data center over the years in response to evolving needs, how do we best accommodate such equipment? Our networks can achieve the same incremental growth at 40% of the expense such growth would incur with past techniques for Clos networks. Further, our approach to designing heterogeneous topologies (i.e., where all the network switches are not identical) achieves 43% higher throughput than a comparable VL2 topology, a heterogeneous network already deployed in Microsoft’s data centers. We acknowledge that the use of random graphs also poses challenges, particularly with regards to efficient routing and physical cabling. We thus present here high-efficiency routing and cabling schemes for such networks as well

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Networks, Uncertainty, Applications and a Crusade for Optimality

In this thesis we address a collection of Network Design problems which are strongly motivated by applications from Telecommunications, Logistics and Bioinformatics. In most cases we justify the need of taking into account uncertainty in some of the problem parameters, and different Robust optimization models are used to hedge against it. Mixed integer linear programming formulations along with sophisticated algorithmic frameworks are designed, implemented and rigorously assessed for the majority of the studied problems. The obtained results yield the following observations: (i) relevant real problems can be effectively represented as (discrete) optimization problems within the framework of network design; (ii) uncertainty can be appropriately incorporated into the decision process if a suitable robust optimization model is considered; (iii) optimal, or nearly optimal, solutions can be obtained for large instances if a tailored algorithm, that exploits the structure of the problem, is designed; (iv) a systematic and rigorous experimental analysis allows to understand both, the characteristics of the obtained (robust) solutions and the behavior of the proposed algorithm