In pursuit of negatively associated measures, this thesis focuses on certain negative correlation properties in matroids. In particular, the results presented contribute to the search for matroids which satisfy $$P(\{X:e,f\in X\}) \leq P(\{X:e\in X\})P(\{X:f\in X\})$$ for certain measures, $P$, on the ground set. Let $\mathcal M$ be a matroid. Let $(y_g:g\in E)$ be a weighting of the ground set and let $${Z = \sum_{X}\left( \prod_{x\in X} y_x\right) }$$ be the polynomial which generates Z-sets, were Z $\in \{$ B,I,S $\}$. For each of these, the sum is over bases, independent sets and spanning sets, respectively. Let $e$ and $f$ be distinct elements of $E$ and let $Z_e$ indicate partial derivative. Then $\mathcal M$ is Z-Rayleigh if $Z_eZ_f-ZZ_{ef}\geq 0$ for every positive evaluation of the $y_g$s. The known elementary results for the B, I and S-Rayleigh properties and two special cases called negative correlation and balance are proved. Furthermore, several new results are discussed. In particular, if a matroid is binary on at most nine elements or paving or rank three, then it is I-Rayleigh if it is B-Rayleigh. Sparse paving matroids are B-Rayleigh. The I-Rayleigh difference for graphs on at most seven vertices is a sum of monomials times squares of polynomials and this same special form holds for all series parallel graphs.

This thesis explores two infinite families of self-complementary arc-transitive graphs: the familiar Paley graphs and the newly discovered Peisert graphs. After studying both families, we examine a result of Peisert which proves the Paley and Peisert graphs are the only self-complementary arc transitive graphs other than one exceptional graph. Then we consider other families of graphs which share many properties with the Paley and Peisert graphs. In particular, we construct an infinite family of self-complementary strongly regular graphs from affine planes. We also investigate the pseudo-Paley graphs of Weng, Qiu, Wang, and Xiang. Finally, we prove a lower bound on the number of maximal cliques of certain pseudo-Paley graphs, thereby distinguishing them from Paley graphs of the same order.

In this thesis we take a specialized approach to edge-colouring by focusing exclusively on multigraphs with high chromatic index. The bulk of our results can be classified into three categories. First, we prove results which aim to characterize those multigraphs achieving known upper bounds. For example, Goldberg's Theorem says that χ'≤ Δ+1+(Δ-2}/(g₀+1) (where χ' denotes chromatic index, Δ denotes maximum degree, and g₀ denotes odd girth). We characterize this bound by proving that for a connected multigraph G, χ'= Δ+1+(Δ-2}/(g₀+1) if and only if G=μC_g₀ and (g₀+1)|2(μ-1) (where μ denotes maximum edge-multiplicity). Our second category of results are new upper bounds for chromatic index in multigraphs, and accompanying polynomial-time edge-colouring algorithms. Our bounds are all approximations to the famous Seymour-Goldberg Conjecture, which asserts that χ'≤ max{⌈ρ⌉, Δ+1} (where ρ=max{(2|E[S]|)/(|S|-1): S⊆V, |S|≥3 and odd}). For example, we refine Goldberg's classical Theorem by proving that χ'≤ max{⌈ρ⌉, Δ+1+(Δ-3)/(g₀+3)}. Our third category of results are characterizations of high chromatic index in general, with particular focus on our approximation results. For example, we completely characterize those multigraphs with χ'> Δ+1+(Δ-3)/(g₀+3). The primary method we use to prove results in this thesis is the method of Tashkinov trees. We first solidify the theory behind this method, and then provide general edge-colouring results depending on Tashkinov trees. We also explore the limits of this method, including the possibility of vertex-colouring graphs which are not line graphs of multigraphs, and the importance of Tashkinov trees with regard to the Seymour-Goldberg Conjecture.

We study techniques, approximation algorithms, structural properties and lower bounds related to applications of linear programs in combinatorial optimization. The following "Steiner tree problem" is central: given a graph with a distinguished subset of required vertices, and costs for each edge, find a minimum-cost subgraph that connects the required vertices. We also investigate the areas of network design, multicommodity flows, and packing/covering integer programs. All of these problems are NP-complete so it is natural to seek approximation algorithms with the best provable approximation ratio. Overall, we show some new techniques that enhance the already-substantial corpus of LP-based approximation methods, and we also look for limitations of these techniques. The first half of the thesis deals with linear programming relaxations for the Steiner tree problem. The crux of our work deals with hypergraphic relaxations obtained via the well-known full component decomposition of Steiner trees; explicitly, in this view the fundamental building blocks are not edges, but hyperedges containing two or more required vertices. We introduce a new hypergraphic LP based on partitions. We show the new LP has the same value as several previously-studied hypergraphic ones; when no Steiner nodes are adjacent, we show that the value of the well-known bidirected cut relaxation is also the same. A new partition uncrossing technique is used to demonstrate these equivalences, and to show that extreme points of the new LP are well-structured. We improve the best known integrality gap on these LPs in some special cases. We show that several approximation algorithms from the literature on Steiner trees can be re-interpreted through linear programs, in particular our hypergraphic relaxation yields a new view of the Robins-Zelikovsky 1.55-approximation algorithm for the Steiner tree problem. The second half of the thesis deals with a variety of fundamental problems in combinatorial optimization. We show how to apply the iterated LP relaxation framework to the problem of multicommodity integral flow in a tree, to get an approximation ratio that is asymptotically optimal in terms of the minimum capacity. Iterated relaxation gives an infeasible solution, so we need to finesse it back to feasibility without losing too much value. Iterated LP relaxation similarly gives an O(k^2)-approximation algorithm for packing integer programs with at most k occurrences of each variable; new LP rounding techniques give a k-approximation algorithm for covering integer programs with at most k variable per constraint. We study extreme points of the standard LP relaxation for the traveling salesperson problem and show that they can be much more complex than was previously known. The k-edge-connected spanning multi-subgraph problem has the same LP and we prove a lower bound and conjecture an upper bound on the approximability of variants of this problem. Finally, we show that for packing/covering integer programs with a bounded number of constraints, for any epsilon > 0, there is an LP with integrality gap at most 1 + epsilon.

The LYM property of a finite standard graded poset is one of the central notions in Sperner theory. It is known that the product of two finite standard graded posets satisfying the LYM properties may not have the LYM property again. In 1974, Harper proved that if two finite standard graded posets satisfying the LYM properties also satisfy rank logarithmic concavities, then their product also satisfies these two properties. However, Harper's proof is rather non-intuitive. Giving a natural proof of Harper's theorem is one of the goals of this thesis. The main new result of this thesis is a characterization of rank-finite standard graded LYM posets that satisfy rank logarithmic concavities. With this characterization theorem, we are able to give a new, natural proof of Harper's theorem. In fact, we prove a strengthened version of Harper's theorem by weakening the finiteness condition to the rank-finiteness condition. We present some interesting applications of the main characterization theorem. We also give a brief history of Sperner theory, and summarize all the ingredients we need for the main theorem and its applications, including a new equivalent condition for the LYM property that is a key for proving our main theorem.

This thesis develops an algorithm for the Schubert calculus of the Grassmanian. Specifically, we state a puzzle-based, synthesis algorithm for a triple intersection of Schubert varieties. Our algorithm is a reformulation of the synthesis algorithm by Bercovici, Collins, Dykema, Li, and Timotin. We replace their combinatorial approach, based on specialized Lebesgue measures, with an approach based on the puzzles of Knutson, Tao and Woodward. The use of puzzles in our algorithm is beneficial for several reasons, foremost among them being the larger body of work exploiting puzzles. To understand the algorithm, we study the necessary Schubert calculus of the Grassmanian to define synthesis. We also discuss the puzzle-based Littlewood-Richardson rule, which connects puzzles to triple intersections of Schubert varieties. We also survey three combinatorial objects related to puzzles in which we include a puzzle-based construction, by King, Tollu, and Toumazet, of the well known Horn inequalities.

This thesis will address several problems in discrete optimization. These problems are considered hard to solve. However, good approximation algorithms for these problems may be helpful in approximating problems in computational biology and computer science. Given an undirected graph G=(V,E) and a family of subsets of vertices S, the minimum crossing spanning tree is a spanning tree where the maximum number of edges crossing any single set in S is minimized, where an edge crosses a set if it has exactly one endpoint in the set. This thesis will present two algorithms for special cases of minimum crossing spanning trees. The first algorithm is for the case where the sets of S are pairwise disjoint. It gives a spanning tree with the maximum crossing of a set being 2OPT+2, where OPT is the maximum crossing for a minimum crossing spanning tree. The second algorithm is for the case where the sets of S form a laminar family. Let b_i be a bound for each S_i in S. If there exists a spanning tree where each set S_i is crossed at most b_i times, the algorithm finds a spanning tree where each set S_i is crossed O(b_i log n) times. From this algorithm, one can get a spanning tree with maximum crossing O(OPT log n). Given an undirected graph G=(V,E), and a family of subsets of vertices S, the minimum crossing perfect matching is a perfect matching where the maximum number of edges crossing any set in S is minimized. A proof will be presented showing that finding a minimum crossing perfect matching is NP-hard, even when the graph is bipartite and the sets of S are pairwise disjoint.

Bilinear pairings on elliptic curves have many cryptographic applications such as identity based encryption, one-round three-party key agreement protocols, and short signature schemes. The elliptic curves which are suitable for pairing-based cryptography are called pairing friendly curves. The prime-order pairing friendly curves with embedding degrees k=3,4 and 6 were characterized by Miyaji, Nakabayashi and Takano. We study this characterization of MNT curves in details. We present explicit algorithms to obtain suitable curve parameters and to construct the corresponding elliptic curves. We also give a heuristic lower bound for the expected number of isogeny classes of MNT curves. Moreover, the related theoretical findings are compared with our experimental results.

In this thesis we consider some network design games that arise from common network design problems. A network design game involves multiple players who control nodes in a network, each of which has a personal interest in seeing their nodes connected in some manner. To this end, the players will submit bids to a mechanism whose task will be to select which of the players to connect, how to connect their nodes, and how much to charge each player for the connection. We rely on many fundamental results from mechanism design (from [8], [9] & [5]) in this thesis and focus our efforts on designing and analyzing cost-sharing methods. That is, for a given set of players and their connection requirements, our algorithms compute a solution that satisfies all the players’ requirements and calculates ’fair’ prices to charge each of them for the connection. Our cost-sharing methods use a primal-dual framework developed by Agrawal, Klein and Ravi in [1] and generalized by Goemans &Williamson in [3]. We modify the algorithms by using the concept of death-time introduced by K¨onemann, Leonardi & Sch¨afer in [6]. Our main result is a 2-budget balanced and cross-monotonic cost sharing method for the downwards monotone set cover game, which arises naturally from any downwards monotone 0, 1-function. We have also designed a 2-budget balanced and cross-monotonic cost sharing method for two versions of the edge cover game arising from the edge cover problem. These games are special cases of the downwards monotone set cover game. By a result by Immorlica, Mahdian & Mirrokni in [4] our result is best possible for the edge cover game. We also designed a cross-monotonic cost sharing method for a network design game we call the Even Parity Connection game arising from the T-Join problem that generalizes proper cut requirement functions. We can show our algorithm returns cost shares that recover at least half the cost of the solution. We conjecture that our cost sharing method for the even parity connection game is competitive and thus 2-budget balance.

Key establishment is the process whereby two or more parties derive a shared secret, typically used for subsequent confidential communication. However, identifying the exact security requirements for key establishment protocols is a non-trivial task. This thesis compares, extends and merges existing security definitions and models for key establishment protocols. The primary focus is on two-party key agreement schemes in the public-key setting. On one hand new protocols are proposed and analyzed in the existing Canetti-Krawzcyk model. On the other hand the thesis develops a security model and novel definition that capture the essential security attributes of the standardized Unified Model key agreement protocol. These analyses lead to the development of a new security model and related definitions that combine and extend the Canetti-Krawzcyk pre- and post- specified peer models in terms of provided security assurances. The thesis also provides a complete analysis of a one-pass key establishment scheme. There are security goals that no one-pass key establishment scheme can achieve, and hence the two-pass security models and definitions need to be adapted for one-pass protocols. The analysis provided here includes the description of the required modification to the underlying security model. Finally, a complete security argument meeting these altered conditions is presented as evidence supporting the security of the one-pass scheme. Lastly, validation and reusing short lived key pairs are related to efficiency, which is a major objective in practice. The thesis considers the formal implication of omitting validation steps and reusing short lived key pairs. The conclusions reached support the generally accepted cryptographic conventions that incoming messages should not be blindly trusted and extra care should be taken when key pairs are reused.