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In the mathematical study of general relativity it is valuable to consider the classification and properties of the metric connection and Riemannian curvature derived from the spacetime metric. The aim of this thesis is to present various results obtained regarding (i) the holonomy groups of simply-connected spacetimes (ii) the problem of deciding if a connection is a metric connection (iii) a sub-class of curvature preserving symmetries and (iv) geodesic preserving symmetries. The holonomy group associated with a spacetime provides a global classification scheme for the metric connection. After the necessary background and definitions, chapter three provides an analysis of the holonomy groups possible in many of the commonly studied classes of spacetime. While every metric is uniquely associated with its metric connection, the converse is not true in general. The question of when a symmetric connection and the curvature derived from it can be associated with a Lorentz signature metric can be addressed via the existence of integrability conditions involving a metric candidate. In chapter four such an analysis is performed in terms of a classification scheme based on the rank and eigenbivector structure of the curvature. The problem is resolved in that the maximum number of such integrability conditions required to guarantee that the connection is a metric connection is given for each (non-trivial) curvature type. Transformations of a spacetime that preserve some geometric quantity or structure are particularly important in general relativity. In chapter five some results regarding transformations that preserve the curvature and its covariant derivatives are presented for spacetimes in which the preservation of the curvature alone is not enough to completely specify the nature of the curvature symmetry.

Optimisation heuristics rely on implicit or explicit assumptions about the structure of the black-box fitness function they optimise. A review of the literature shows that understanding of structure and linkage is helpful to the design and analysis of heuristics. The aim of this thesis is to investigate the role that problem structure plays in heuristic optimisation. Many heuristics use ordinal operators; which are those that are invariant under monotonic transformations of the fitness function. In this thesis we develop a classification of pseudo-Boolean functions based on rank-invariance. This approach classifies functions which are monotonic transformations of one another as equivalent, and so partitions an infinite set of functions into a finite set of classes. Reasoning about heuristics composed of ordinal operators is, by construction, invariant over these classes. We perform a complete analysis of 2-bit and 3-bit pseudo-Boolean functions. We use Walsh analysis to define concepts of necessary, unnecessary, and conditionally necessary interactions, and of Walsh families. This helps to make precise some existing ideas in the literature such as benign interactions. Many algorithms are invariant under the classes we define, which allows us to examine the difficulty of pseudo-Boolean functions in terms of function classes. We analyse a range of ordinal selection operators for an EDA. Using a concept of directed ordinal linkage, we define precedence networks and precedence profiles to represent key algorithmic steps and their interdependency in terms of problem structure. The precedence profiles provide a measure of problem difficulty. This corresponds to problem difficulty and algorithmic steps for optimisation. This work develops insight into the relationship between function structure and problem difficulty for optimisation, which may be used to direct the development of novel algorithms. Concepts of structure are also used to construct easy and hard problems for a hill-climber.