In this thesis we prove intractability results for well studied problems in computational learning and approximation. Let ε , mu > 0 be arbitrarily small constants and t be an arbitrary constant positive integer. We show an almost optimal hardness factor of d[superscript{1-ε}] for computing an equivalent DNF expression with minimum terms for a boolean function on d variables, given its truth table. In the study of weak learnability, we prove an optimal 1/2 + ε inapproximability for the accuracy of learning an intersection of two halfspaces with an intersection of t halfspaces. Further, we study the learnability of small DNF formulas, and prove optimal 1/2 + ε inapproximability for the accuracy of learning (i) a two term DNF by a t term DNF, and (ii) an AND under adversarial mu-noise by a t-CNF. In addition, we show a 1 - 2[superscript{-d}] + ε inapproximability for accurately learning parities (over GF(2)), under adversarial mu-noise, by degree d polynomials, where d is a constant positive integer. We also provide negative answers to the possibility of stronger semi-definite programming (SDP) relaxations yielding much better approximations for graph partitioning problems such as Maximum Cut and Sparsest Cut by constructing integrality gap examples for them. For Maximum Cut and Sparsest Cut we construct examples -- with gaps alpha[superscript{-1}] - ε (alpha is the Goemans-Williamson constant) and Omega((logloglog n)[superscript{1/13}]) respectively -- for the standard SDP relaxations augmented with O((logloglog n)[superscript{1/6}]) rounds of Sherali-Adams constraints. The construction for Sparsest Cut also implies that an n-point negative type metric may incur a distortion of Omega((logloglog n)[superscript{1/ 13}]) to embed into ell_1 even if the induced submetric on every subset of O((logloglog n)[superscript{1/6}]) points is isometric to ell_1. We also construct an integrality gap of Omega(loglog n) for the SDP relaxation for Uniform Sparsest Cut problem augmented with triangle inequalities, disproving a well known conjecture of Arora, Rao and Vazirani.

There are many times in which a critical choice between proposed system architectures must be made. Two situations in particular motivate this dissertation: a "Cambrian explosion" when no dominant rchitecture has arisen, and times in which developments enable challenges to a dominant incumbent. In each situation, the advance of core technologies is key. This dissertation features a new computing technique to systematically explore the interaction of technological progress with architectural choices. This technique is founded upon a graph theoretic formulation of architecture, which enables the consideration of multifunctional components and modularity v. synergy trades. The technique utilizes a genetic algorithm formulated for graphs, and a solver that automatically constrains and optimizes component design variables. The use of quantitative technology models, graph theoretic formulation, and optimization algorithms together enables a systematic exploration of both time and combinatorial spaces. The quantitative results of this exploration enhance the strategic view of technology planners.

Combinatorial optimisation problems (COPs) pervade human society: scheduling, design, layout, distribution, timetabling, resource allocation and project management all feature problems where the solution is some combination of elements, the overall value of which needs to be either maximised or minimised (i.e., optimised), typically subject to a number of constraints. Thus, techniques to efficiently solve such problems are an important area of research. A popular group of optimisation algorithms are the metaheuristics, approaches that specify how to search the space of solutions in a problem independent way so that high quality solutions are likely to result in a reasonable amount of computational time. Although metaheuristic algorithms are specified in a problem independent manner, they must be tailored to suit each particular problem to which they are applied. This thesis investigates a number of aspects of the application of the relatively new Ant Colony Optimisation (ACO) metaheuristic to different COPs.The standard ACO metaheuristic is a constructive algorithm loosely based on the foraging behaviour of ant colonies, which are able to find the shortest path to a food source by indirect communication through pheromones. ACO’s artificial pheromone represents a model of the solution components that its artificial ants use to construct solutions. Developing an appropriate pheromone representation is a key aspect of the application of ACO to a problem. An examination of existing ACO applications and the constructive approach more generally reveals how the metaheuristic can be applied more systematically across a range of COPs. The two main issues addressed in this thesis are biases inherent in the constructive process and the systematic selection of pheromone representations.The systematisation of ACO should lead to more consistently high performance of the algorithm across different problems. Additionally, it supports the creation of a generalised ACO system, capable of adapting itself to suit many different combinatorial problems without the need for manual intervention.

This thesis presents and develops a theory of self-reduction. This process is used to map instances of combinatorial optimisation problems onto smaller, more easily solvable instances in such a way that a solution of the former can be readily re-constructed, without loss of information or quality, from a solution of the latter. Self-reduction rules are surveyed for the Graph Colouring Problem, the Maximum Clique Problem, the Steiner Problem in Graphs, the Bin Packing Problem and the Set Covering Problem. This thesis introduces the problem of determining the maximum sequence of self-reductions on a given structure, and shows how the theory of confluence can be adapted from term re-writing to solve this problem by identifying rule sets for which all maximal reduction sequences are equivalent. Such confluence results are given for a number of reduction rules on problems on discrete systems. In contrast, NP-hardness results are also presented for some reduction rules. A probabilistic analysis of self-reductions on graphs is performed, showing that the expected number of self-reductions on a graph tends to zero as the order of the graph tends to infinity. An empirical study is performed comparing the performance of self-reduction, graph decomposition and direct methods of solving the Graph Colouring and Set Covering Problems. The results show that self-reduction is a potentially valuable, but sometimes erratic, method of finding exact solutions to combinatorial problems.

Thesis (Ph. D.)--State University of New York at Binghamton, Department of Computer Science, Thomas J. Watson School of Engineering and Applied Science, 2007. / Includes bibliographical references.

Thesis (Ph. D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Thomas, Robin; Committee Co-Chair: Cook, William J.; Committee Member: Dvorak, Zdenek; Committee Member: Parker, Robert G.; Committee Member: Yu, Xingxing.

Thesis (M.S.)--University of Texas at El Paso, 2008. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.

Thesis (Ph. D.)--University of Sydney, 2009. / Title from title screen (viewed May 1, 2009) Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy Economics to the Faculty of Economics and Business, University of Sydney. Degree awarded 2009; thesis submitted 2008. Bibliography: leaves 93-97. Also available in print form.

Thesis (Ph.D.)--Computing, Georgia Institute of Technology, 2009. / Committee Chair: Bader, David; Committee Member: Berry, Jonathan; Committee Member: Fujimoto, Richard; Committee Member: Saini, Subhash; Committee Member: Vuduc, Richard

Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Trotter, William T.; Committee Member: Duke, Richard A.; Committee Member: Randall, Dana; Committee Member: Thomas, Robin; Committee Member: Yu, Xingxing