For many combinatorial objects we can associate a natural probability distribution on the members of the class, and we can then call the resulting class a class of random structures. Random structures form good models of many real world problems, in particular real networks and disordered media. For many such problems, the systems under consideration can be very large, and we often care about whether a property holds most of the time. In particular, for a given class of random structures, we say that a property holds with high probability if the probability that that property holds tends to one as the size of the structures increase. We examine several classes of random structures with real world applications, and look at some properties of each that hold with high probability. First we look at percolation in 3 dimensional lattices, giving a method for producing rigorous confidence intervals on the percolation threshold. Next we look at random geometric graphs, first examining the connectivity thresholds of nearest neighbour models, giving good bounds on the threshold for a new variation on these models useful for modelling wireless networks, and then look at the cop number of the Gilbert model. Finally we look at the structure of random sum-free sets, in particular examining what the possible densities of such sets are, what substructures they can contain, and what superstructures they belong to.

In 2003, Gomory and Johnson gave two different three-slope T-space facet constructions, both of which shared a slope with the corresponding Gomory mixed-integer cut. We give a new three-slope facet which is independent of the GMIC and also give a four-slope T-space facet construction, which to our knowledge, is the first four-slope construction. We describe an enumerative framework for the discovery of T-space facets. Using an algorithm by Harvey for computing integer hulls in the plane, we give a heuristic for quickly computing lattice-free triangles. Given two rows of the tableau, we derive how to exactly calculate lattice-free triangles and quadrilaterals in the plane which can be used to derive facet-defining inequalities of the integer hull. We then present computational results using these derivations where non-basic integer variables are strengthened using Balas-Jeroslow lifting.

Union-free families of subsets of [n] = {1, . . ., n} have been studied in Frankl and Füredi (Eur J Combin 5:127-131, 1984). In this paper, we provide a complete characterization of maximal symmetric difference-free families of subsets of [n].

Any reasonably large group of individuals, families, states, and parties exhibits the phenomenon of subgroup formations within the group such that the members of each group have a strong connection or bonding between each other. The reasons of the formation of these subgroups that we call alliances differ in different situations, such as, kinship and friendship (in the case of individuals), common economic interests (for both individuals and states), common political interests, and geographical proximity. This structure of alliances is not only prevalent in social networks, but it is also an important characteristic of similarity networks of natural and unnatural objects. (A similarity network defines the links between two objects based on their similarities). Discovery of such structure in a data set is called clustering or unsupervised learning and the ability to do it automatically is desirable for many applications in the areas of pattern recognition, computer vision, artificial intelligence, behavioral and social sciences, life sciences, earth sciences, medicine, and information theory. In this dissertation, we study a graph theoretical model of alliances where an alliance of the vertices of a graph is a set of vertices in the graph, such that every vertex in the set is adjacent to equal or more vertices inside the set than the vertices outside it. We study the problem of partitioning a graph into alliances and identify classes of graphs that have such a partition. We present results on the relationship between the existence of such a partition and other well known graph parameters, such as connectivity, subgraph structure, and degrees of vertices. We also present results on the computational complexity of finding such a partition. An alliance cover set is a set of vertices in a graph that contains at least one vertex from every alliance of the graph. The complement of an alliance cover set is an alliance free set, that is, a set that does not contain any alliance as a subset. We study the properties of these sets and present tight bounds on their cardinalities. In addition, we also characterize the graphs that can be partitioned into alliance free and alliance cover sets. Finally, we present an approximate algorithm to discover alliances in a given graph. At each step, the algorithm finds a partition of the vertices into two alliances such that the alliances are strongest among all such partitions. The strength of an alliance is defined as a real number p, such that every vertex in the alliance has at least p times more neighbors in the set than its total number of neighbors in the graph). We evaluate the performance of the proposed algorithm on standard data sets.