A Note on the Art of Network Design Problems

In this study, we describe some Network Design Problems (NDPs) as well as the network flowbased improvement algorithm for neighbourhood search defined by cycles. The main part of the study is structured around the formulation of the expected duration of stay in the educational system as a NDP. The fundamental matrix of the absorbing Markov chain is employed in computing the expected duration of each flow in the system. We shall illustrate the new graphtheoretic formulation for the educational system using datasets from a university setting. The paper concludes with suggestions for future directions of research.Keywords: absorbing Markov chain; educational system; graph theory; network design

Valid Inequalities and Facets for Multi-Module (Survivable) Capacitated Network Design Problem

In this dissertation, we develop new methodologies and algorithms to solve the multi-module (survivable) network design problem. Many real-world decision-making problems can be modeled as network design problems, especially on networks with capacity requirements on arcs or edges. In most cases, network design problems of this type that have been studied involve different types of capacity sizes (modules), and we call them the multi-module capacitated network design (MMND) problem. MMND problems arise in various industrial applications, such as transportation, telecommunication, power grid, data centers, and oil production, among many others. In the first part of the dissertation, we study the polyhedral structure of the MMND problem. We summarize current literature on polyhedral study of MMND, which generates the family of the so-called cutset inequalities based on the traditional mixed integer rounding (MIR). We then introduce a new family of inequalities for MMND based on the so-called n-step MIR, and show that various classes of cutset inequalities in the literature are special cases of these inequalities. We do so by studying a mixed integer set, the cutset polyhedron, which is closely related to MMND. We We also study the strength of this family of inequalities by providing some facet-defining conditions. These inequalities are then tested on MMND instances, and our computational results show that these classes of inequalities are very effective for solving MMND problems. Generalizations of these inequalities for some variants of MMND are also discussed. Network design problems have many generalizations depending on the application. In the second part of the dissertation, we study a highly applicable form of SND, referred to as multi-module SND (MM-SND), in which transmission capacities on edges can be sum of integer multiples of differently sized capacity modules. For the first time, we formulate MM-SND as a mixed integer program (MIP) using preconfigured-cycles (p-cycles) to reroute flow on failed edges. We derive several classes of valid inequalities for this MIP, and show that the valid inequalities previously developed in the literature for single-module SND are special cases of our inequalities. Furthermore, we show that our valid inequalities are facet-defining for MM-SND in many cases. Our computational results, using a heuristic separation algorithm, show that these inequalities are very effective in solving MM-SND. In particular they are more effective than compared to using single-module inequalities alone. Lastly, we generalize the inequalities for MMND for other mixed integer sets relaxed from MMND and the cutset polyhedron. These inequalities also generalize several valid inequalities in the literature. We conclude the dissertation by summarizing the work and pointing out potential directions for future research

Approximation Complexity of Optimization Problems : Structural Foundations and Steiner Tree Problems

In this thesis we study the approximation complexity of the Steiner Tree Problem and related problems as well as foundations in structural complexity theory. The Steiner Tree Problem is one of the most fundamental problems in combinatorial optimization. It asks for a shortest connection of a given set of points in an edge-weighted graph. This problem and its numerous variants have applications ranging from electrical engineering, VLSI design and transportation networks to internet routing. It is closely connected to the famous Traveling Salesman Problem and serves as a benchmark problem for approximation algorithms. We give a survey on the Steiner tree Problem, obtaining lower bounds for approximability of the (1,2)-Steiner Tree Problem by combining hardness results of Berman and Karpinski with reduction methods of Bern and Plassmann. We present approximation algorithms for the Steiner Forest Problem in graphs and bounded hypergraphs, the Prize Collecting Steiner Tree Problem and related problems where prizes are given for pairs of terminals. These results are based on the Primal-Dual method and the Local Ratio framework of Bar-Yehuda. We study the Steiner Network Problem and obtain combinatorial approximation algorithms with reasonable running time for two special cases, namely the Uniform Uncapacitated Case and the Prize Collecting Uniform Uncapacitated Case. For the general case, Jain's algorithms obtains an approximation ratio of 2, based on the Ellipsoid Method. We obtain polynomial time approximation schemes for the Dense Prize Collecting Steiner Tree Problem, Dense k-Steiner Problem and the Dense Class Steiner Tree Problem based on the methods of Karpinski and Zelikovsky for approximating the Dense Steiner Tree Problem. Motivated by the question which parameters make the Steiner Tree problem hard to solve, we make an excurs into Fixed Parameter Complexity, focussing on structural aspects of the W-Hierarchy. We prove a Speedup Theorem for the classes FPT and SP and versions if Levin's Lower Bound Theorem for the class SP as well as for Randomized Space Complexity. Starting from the approximation schemes for the dense Steiner Tree problems, we deal with the efficiency of polynomial time approximation schemes in general. We separate the class EPTAS from PTAS under some reasonable complexity theoretic assumption. The same separation was achieved by Cesaty and Trevisan under some assumtion from Fixed Parameter Complexity. We construct an oracle under which our assumtion holds but that of Cesati and Trevisan does not, which implies that using relativizing proof techniques one cannot show that our assumption implies theirs

Approximation Algorithms for a Capacitated Network Design Problem

We study a capacitated network design problem with applications in local access network design. Given a network, the problem is to route flow from several sources to a sink and to install capacity on the edges to support the flow at minimum cost. Capacity can be purchased only in multiples of a fixed quantity. All the flow from a source must be routed in a single path to the sink. This NP-hard problem generalizes the Steiner tree problem and also more effectively models the applications traditionally formulated as capacitated tree problems. We present an approximation algorithm with performance ratio (ρST + 2) where ρST is the performance ratio of any approximation algorithm for the minimum Steiner tree problem. When all sources have unit demand, the ratio improves to ρST + 1) and, in particular, to 2 when all nodes in the graph are sources