Consider a probability distribution on the permutations of n elements. If the probability of each permutation is proportional to θK, where K is the number of cycles in the permutation, then we say that the distribution generates a θ-biased random permutation. A random permutation is a special θ-biased random permutation with θ = 1. The mth moment of the rth longest cycle of a random permutation is Θ(nm), regardless of r and θ. The joint moments are derived, and it is shown that the longest cycles of a permutation can either be positively or negatively correlated, depending on θ. The mth moments of the rth shortest cycle of a random permutation is Θ(nm−θ/(ln n)r−1) when θ < m, Θ((ln n)r) when θ = m, and Θ(1) when θ > m. The exponent of cycle lengths at the 100qth percentile goes to q with zero variance. The exponent of the expected cycle lengths at the 100qth percentile is at least q due to the Jensen’s inequality, and the exact value is derived.

A k-net of order n is an incidence structure consisting of n2 points and nk lines. Two lines are said to be parallel if they do not intersect. A k-net of order n satisfies the following four axioms: (i) every line contains n points; (ii) parallelism is an equivalence relation on the set of lines; (iii) there are k parallel classes, each consisting of n lines and (iv) any two non-parallel lines meet exactly once. A Latin square of order n is an n by n array of symbols in which each row and column contains each symbol exactly once. Two Latin squares L and M are said to be orthogonal if the n2 ordered pairs (Li,j , Mi,j ) are all distinct. A set of t mutual ly orthogonal Latin squares is a collection of Latin squares, necessarily of the same order, that are pairwise orthogonal. A k-net of order n is combinatorially equivalent to k − 2 mutually orthogonal Latin squares of order n. It is this equivalence that motivates much of the work in this thesis. One of the most important open questions in the study of Latin squares is: given an order n what is the maximum number of mutually orthogonal Latin squares of that order? This is a particularly interesting question when n is congruent to two modulo four. A code is constructed from a net by defining the characteristic vectors of lines to be generators of the code over the finite field F2 . Codes allow the structure of nets to be profitably explored using techniques from linear algebra. In this dissertation a framework is developed to study linear dependence in the code of the net N6 of order ten. A complete classification and combinatorial description of such dependencies is given. This classification could facilitate a computer search for a net or could be used in conjunction with more refined techniques to rule out the existence of these nets combinatorially. In more generality relations in 4-nets of order congruent to two modulo four are also characterized. One type of dependency determined algebraically is shown not to be combinatorially feasible in a net N6 of order ten. Some dependencies are shown to be related geometrically, allowing for a concise classification. Using a modification of the dimension argument first introduced by Dougherty [19] new upper bounds are established on the dimension of codes of nets of order congruent to two modulo four. New lower bounds on some of these dimensions are found using a combinatorial argument. Certain constraints on the dimension of a code of a net are shown to imply the existence of specific combinatorial structures in the net. The problem of packing points into lines in a prescribed way is related to packing problems in graphs and more general packing problems in combinatorics. This dissertation exploits the geometry of nets and symmetry of complete multipartite graphs and combinatorial designs to further unify these concepts in the context of the problems studied here.

Graph searching problems are described as games played on graphs, between a set of searchers and a fugitive. Variants of the game restrict the abilities of the searchers and the fugitive and the corresponding search number (the least number of searchers that have a winning strategy) is related to several well-known parameters in graph theory. One popular variant is called the Cops and Robber game, where the searchers (cops) and the fugitive (robber) move in rounds, and in each round they move to an adjacent vertex. This game, defined in late 1970's, has been studied intensively. The most famous open problem is Meyniel's conjecture, which states that the cop number (the minimum number of cops that can always capture the robber) of a connected graph on n vertices is O(sqrt n). We consider a version of the Cops and Robber game, where the robber is faster than the cops, but is not allowed to jump over the cops. This version was first studied in 2008. We show that when the robber has speed s, the cop number of a connected n-vertex graph can be as large as Omega(n^(s/s+1)). This improves the Omega(n^(s-3/s-2)) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, to appear). We also conjecture a general upper bound O(n^(s/s+1)) for the cop number, generalizing Meyniel's conjecture. Then we focus on the version where the robber is infinitely fast, but is again not allowed to jump over the cops. We give a mathematical characterization for graphs with cop number one. For a graph with treewidth tw and maximum degree Delta, we prove the cop number is between (tw+1)/(Delta+1) and tw+1. Using this we show that the cop number of the m-dimensional hypercube is between c1 n / m sqrt(m) and c2 n / m for some constants c1 and c2. If G is a connected interval graph on n vertices, then we give a polynomial time 3-approximation algorithm for finding the cop number of G, and prove that the cop number is O(sqrt(n)). We prove that given n, there exists a connected chordal graph on n vertices with cop number Omega(n/log n). We show a lower bound for the cop numbers of expander graphs, and use this to prove that the random G(n,p) that is not very sparse, asymptotically almost surely has cop number between d1 / p and d2 log (np) / p for suitable constants d1 and d2. Moreover, we prove that a fixed-degree regular random graph with n vertices asymptotically almost surely has cop number Theta(n).

A clique of a graph G is a set of pairwise adjacent nodes of G. Similarly, a biclique (U, V ) of a bipartite graph G is a pair of disjoint, independent vertex sets such that each node in U is adjacent to every node in V in G. We consider the problems of identifying the maximum clique of a graph, known as the maximum clique problem, and identifying the biclique (U, V ) of a bipartite graph that maximizes the product |U | · |V |, known as the maximum edge biclique problem. We show that finding a clique or biclique of a given size in a graph is equivalent to finding a rank one matrix satisfying a particular set of linear constraints. These problems can be formulated as rank minimization problems and relaxed to convex programming by replacing rank with its convex envelope, the nuclear norm. Both problems are NP-hard yet we show that our relaxation is exact in the case that the input graph contains a large clique or biclique plus additional nodes and edges. For each problem, we provide two analyses of when our relaxation is exact. In the first, the diversionary edges are added deterministically by an adversary. In the second, each potential edge is added to the graph independently at random with fixed probability p. In the random case, our bounds match the earlier bounds of Alon, Krivelevich, and Sudakov, as well as Feige and Krauthgamer for the maximum clique problem. We extend these results and techniques to the k-disjoint-clique problem. The maximum node k-disjoint-clique problem is to find a set of k disjoint cliques of a given input graph containing the maximum number of nodes. Given input graph G and nonnegative edge weights w, the maximum mean weight k-disjoint-clique problem seeks to identify the set of k disjoint cliques of G that maximizes the sum of the average weights of the edges, with respect to w, of the complete subgraphs of G induced by the cliques. These problems may be considered as a way to pose the clustering problem. In clustering, one wants to partition a given data set so that the data items in each partition or cluster are similar and the items in different clusters are dissimilar. For the graph G such that the set of nodes represents a given data set and any two nodes are adjacent if and only if the corresponding items are similar, clustering the data into k disjoint clusters is equivalent to partitioning G into k-disjoint cliques. Similarly, given a complete graph with nodes corresponding to a given data set and edge weights indicating similarity between each pair of items, the data may be clustered by solving the maximum mean weight k-disjoint-clique problem. We show that both instances of the k-disjoint-clique problem can be formulated as rank constrained optimization problems and relaxed to semidefinite programs using the nuclear norm relaxation of rank. We also show that when the input instance corresponds to a collection of k disjoint planted cliques plus additional edges and nodes, this semidefinite relaxation is exact for both problems. We provide theoretical bounds that guarantee exactness of our relaxation and provide empirical examples of successful applications of our algorithm to synthetic data sets, as well as data sets from clustering applications.

For a given graph, G, the crossing number crₐ(G) denotes the minimum number of edge crossings when a graph is drawn on an orientable surface of genus a. The sequence cr₀(G), cr₁(G), ... is said to be the crossing sequence of a G. An equivalent definition exists for non-orientable surfaces. In 1983, Jozef Širáň proved that for every decreasing, convex sequence of non-negative integers, there is a graph G such that this sequence is the crossing sequence of G. This main result of this thesis proves the existence of a graph with non-convex crossing sequence of arbitrary length.

We introduce methods for dealing with linear programming (LP) problems with uncertain data, using the notion of weighted analytic centers. Our methods are based on high interaction with the decision maker (DM) and try to find solutions which satisfy most of his/her important criteria/goals. Starting with the drawbacks of different methods for dealing with uncertainty in LP, we explain how our methods improve most of them. We prove that, besides many practical advantages, our approach is theoretically as strong as robust optimization. Interactive cutting-plane algorithms are developed for concave and quasi-concave utility functions. We present some probabilistic bounds for feasibility and evaluate our approach by means of computational experiments.

In this thesis, we provide a summary of fully homomorphic encryption, and in particular, look at the BGV encryption scheme by Brakerski, Gentry, and Vaikuntanathan; as well the DGHV encryption scheme by van Dijk, Gentry, Halevi, and Vaikuntanathan. We explain the mechanisms developed by Gentry in his breakthrough work, and show examples of how they are used. While looking at the BGV encryption scheme, we make improvements to the underlying lemmas dealing with modulus switching and noise management, and show that the lemmas as currently stated are false. We then examine a lower bound on the hardness of the Learning With Errors lattice problem, and use this to develop specific parameters for the BGV encryption scheme at a variety of security levels. We then study the DGHV encryption scheme, and show how the somewhat homomorphic encryption scheme can be implemented as both a fully homomorphic encryption scheme with bootstrapping, as well as a leveled fully homomorphic encryption scheme using the techniques from the BGV encryption scheme. We then extend the parameters from the optimized version of this scheme to higher security levels, and describe a more straightforward way of arriving at these parameters.

In elliptic curve cryptography, cryptosystems are based on an additive subgroup of an elliptic curve defined over a finite field, and the hardness of the Elliptic Curve Discrete Logarithm Problem is dependent on the order of this subgroup. In particular, we often want to find a subgroup with large prime order. Hence when finding a suitable curve for cryptography, counting the number of points on the curve is an essential step in determining its security. In 1985, René Schoof proposed the first deterministic polynomial-time algorithm for point counting on elliptic curves over finite fields. The algorithm was improved by Noam Elkies and Oliver Atkin, resulting in an algorithm which is sufficiently fast for practical purposes. The enhancements leveraged the arithmetic properties of the l-th classical modular polynomial, where l- is either an Elkies or Atkin prime. As the Match-Sort algorithm relating to Atkin primes runs in exponential time, it is eschewed in common practice. In this thesis, I will discuss my implementation of the Schoof-Elkies-Atkin algorithm in C++, which makes use of the NTL package. The implementation also supports the computation of classical modular polynomials via isogeny volcanoes, based on the methods proposed recently by Bröker, Lauter and Sutherland. Existing complexity analysis of the Schoof-Elkies-Atkin algorithm focuses on its asymptotic performance. As such, there is no estimate of the actual impact of the Match-Sort algorithm on the running time of the Schoof-Elkies-Atkin algorithm for elliptic curves defined over prime fields of cryptographic sizes. I will provide rudimentary estimates for the largest Elkies or Atkin prime used, and discuss the variants of the Schoof-Elkies-Atkin algorithm using their run-time performances. The running times of the SEA variants supports the use Atkin primes for prime fields of sizes up to 256 bits. At this size, the selective use of Atkin primes runs in half the time of the Elkies-only variant on average. This suggests that Atkin primes should be used in point counting on elliptic curves of cryptographic sizes.

Dynamic Programming is an optimization technique used in computer science and mathematics. Introduced in the 1950s, it has been applied to many classic combinatorial optimization problems, such as the Shortest Path Problem, the Knapsack Problem, and the Traveling Salesman Problem, with varying degrees of practical success. In this thesis, we present two applications of dynamic programming to optimization problems. The first application is as a method to compute the Branch-Cut-and-Price (BCP) family of lower bounds for the Traveling Salesman Problem (TSP), and several vehicle routing problems that generalize it. We then prove that the BCP family provides a set of lower bounds that is at least as strong as the Approximate Linear Program (ALP) family of lower bounds for the TSP. The second application is a novel dynamic programming model used to determine the placement of cuts for a particular form of skull surgery called Cranial Vault Remodeling.

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