The capacitated facility location problem is a well known problem in combinatorial optimization and operations research. In it, we are given a set of clients and a set of possible facility locations. Each client has a certain demand that needs to be satisfied from open facilities, without exceeding their capacity. Whenever we open a facility we incur in a corresponding opening cost. Whenever demand is served, we incur in an assignment cost; depending on the distance the demand travels. The goal is to open a set of facilities that satisfy all demands while minimizing the total opening and assignment costs. In this thesis, we present two novel LP-based approximation algorithms for the capacitated facility location problem. The first algorithm is based on LP-rounding techniques, and is designed for the special case of the capacitated facility location problem where capacities are uniform and assignment costs are given by a tree metric. The second algorithm follows a primal-dual approach, and works for the general case. For both algorithms, we obtain an approximation guarantee that is linear on the size of the problem. To the best of our knowledge, there are no LP-based algorithms known, for the type of instances that we focus on, that achieve a better performance.

Recent work on many problems in additive combinatorics, such as Roth's Theorem, has shown the usefulness of first studying the problem in a finite field environment. Using the techniques of Bourgain to give a result in other settings such as general abelian groups, the author gives a walk through, including proof, of Roth's theorem in both the one dimensional and two dimensional cases (it would be more accurate to refer to the two dimensional case as Shkredov's Theorem). In the one dimensional case the argument is at its base Meshulam's but the structure will be essentially Green's. Let Ϝⁿ [subscript p], p ≠ 2 be the finite field of cardinality N = pⁿ. For large N, any subset A ⊂ Ϝⁿ [subscript p] of cardinality ∣A ∣≳ N ∕ log N must contain a triple of the form {x, x + d, x + 2d} for x, d ∈ Ϝⁿ [subscript p], d ≠ 0. In the two dimensional case the argument is Lacey and McClain who made considerable refinements to this argument of Green who was bringing the argument to the finite field case from a paper of Shkredov. Let Ϝ ⁿ ₂ be the finite field of cardinality N = 2ⁿ. For all large N, any subset A⊂ Ϝⁿ ₂ × Ϝⁿ ₂ of cardinality ∣A ∣≳ N ² (log n) − [superscript epsilon], ε <, 1, must contain a corner {(x, y), (x + d, y), (x, y + d)} for x, y, d ∈ Ϝⁿ₂ and d ≠ 0.

We construct a new operation among representations of the symmetric group that interpolates between the classical internal and external products, which are defined in terms of tensor product and induction of representations. Following Malvenuto and Reutenauer, we pass from symmetric functions to non-commutative symmetric functions and from there to the algebra of permutations in order to relate the internal and external products to the composition and convolution of linear endomorphisms of the tensor algebra. The new product we construct corresponds to the Heisenberg product of endomorphisms of the tensor algebra. For symmetric functions, the Heisenberg product is given by a construction which combines induction and restriction of representations. For non-commutative symmetric functions, the structure constants of the Heisenberg product are given by an explicit combinatorial rule which extends a well-known result of Garsia, Remmel, Reutenauer, and Solomon for the descent algebra. We describe the dual operation among quasi-symmetric functions in terms of alphabets.

The Time-Dependent Traveling Salesman Problem is a generalization of the well-known Traveling Salesman Problem, where the cost for travel between two nodes is dependent on the nodes and their position in the tour. Inequalities for the Asymmetric TSP can be easily extended to the TDTSP, but the added time information can be used to strengthen these inequalities. We look at extending the Lifted Cycle Inequalities, a large family of inequalities for the ATSP. We define a new inequality, the Extended Cycle (X-cycle) Inequality, based on cycles in the graph. We extend the results of Balas and Fischetti for Lifted Cycle Inequalities to define Lifted X-cycle Inequalities. We show that the Lifted X-cycle Inequalities include some inequalities which define facets of the submissive of the TDTS Polytope.

An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves efficiently. For ordinary curves of the same endomorphism ring, the previous fastest algorithm known has a worst case running time which is exponential in the length of the input. In this thesis we solve this problem in subexponential time under reasonable heuristics. We give two versions of our algorithm, a slower version assuming GRH and a faster version assuming stronger heuristics. Our approach is based on factoring the ideal corresponding to the kernel of the isogeny, modulo principal ideals, into a product of smaller prime ideals for which the isogenies can be computed directly. Combined with previous work of Bostan et al., our algorithm yields equations for large degree isogenies in quasi-optimal time given only the starting curve and the kernel.

The Lovasz theta function and the associated convex sets known as theta bodies are fundamental objects in combinatorial and semidefinite optimization. They are accompanied by a rich duality theory and deep connections to the geometric concept of orthonormal representations of graphs. In this thesis, we investigate several ramifications of the theory underlying these objects, including those arising from the illuminating viewpoint of duality. We study some optimization problems over unit-distance representations of graphs, which are intimately related to the Lovasz theta function and orthonormal representations. We also strengthen some known results about dual descriptions of theta bodies and their variants. Our main goal throughout the thesis is to lay some of the foundations for using semidefinite optimization and convex analysis in a way analogous to how polyhedral combinatorics has been using linear optimization to prove min-max theorems. A unit-distance representation of a graph $G$ maps its nodes to some Euclidean space so that adjacent nodes are sent to pairs of points at distance one. The hypersphere number of $G$, denoted by $t(G)$, is the (square of the) minimum radius of a hypersphere that contains a unit-distance representation of $G$. Lovasz proved a min-max relation describing $t(G)$ as a function of $\vartheta(\overline{G})$, the theta number of the complement of $G$. This relation provides a dictionary between unit-distance representations in hyperspheres and orthonormal representations, which we exploit in a number of ways: we develop a weighted generalization of $t(G)$, parallel to the weighted version of $\vartheta$; we prove that $t(G)$ is equal to the (square of the) minimum radius of an Euclidean ball that contains a unit-distance representation of $G$; we abstract some properties of $\vartheta$ that yield the famous Sandwich Theorem and use them to define another weighted generalization of $t(G)$, called ellipsoidal number of $G$, where the unit-distance representation of $G$ is required to be in an ellipsoid of a given shape with minimum volume. We determine an analytic formula for the ellipsoidal number of the complete graph on $n$ nodes whenever there exists a Hadamard matrix of order $n$. We then study several duality aspects of the description of the theta body $\operatorname{TH}(G)$. For a graph $G$, the convex corner $\operatorname{TH}(G)$ is known to be the projection of a certain convex set, denoted by $\widehat{\operatorname{TH}}(G)$, which lies in a much higher-dimensional matrix space. We prove that the vertices of $\widehat{\operatorname{TH}}(G)$ are precisely the symmetric tensors of incidence vectors of stable sets in $G$, thus broadly generalizing previous results about vertices of the elliptope due to Laurent and Poljak from 1995. Along the way, we also identify all the vertices of several variants of $\widehat{\operatorname{TH}}(G)$ and of the elliptope. Next we introduce an axiomatic framework for studying generalized theta bodies, based on the concept of diagonally scaling invariant cones, which allows us to prove in a unified way several characterizations of $\vartheta$ and the variants $\vartheta'$ and $\vartheta^+$, introduced independently by Schrijver, and by McEliece, Rodemich, and Rumsey in the late 1970's, and by Szegedy in 1994. The beautiful duality equation which states that the antiblocker of $\operatorname{TH}(G)$ is $\operatorname{TH}(\overline{G})$ is extended to this setting. The framework allows us to treat the stable set polytope and its classical polyhedral relaxations as generalized theta bodies, using the completely positive cone and its dual, and it allows us to derive a (weighted generalization of a) copositive formulation for the fractional chromatic number due to Dukanovic and Rendl in 2010 from a completely positive formulation for the stability number due to de Klerk and Pasechnik in 2002. Finally, we study a non-convex constraint for semidefinite programs (SDPs) that may be regarded as analogous to the usual integrality constraint for linear programs. When applied to certain classical SDPs, it specializes to the standard rank-one constraint. More importantly, the non-convex constraint also applies to the dual SDP, and for a certain SDP formulation of $\vartheta$, the modified dual yields precisely the clique covering number. This opens the way to study some exactness properties of SDP relaxations for combinatorial optimization problems akin to the corresponding classical notions from polyhedral combinatorics, as well as approximation algorithms based on SDP relaxations.

In this thesis, we show that several natural questions about automatic sequences can be expressed as logical predicates and then decided mechanically. We extend known results in this area to broader classes of sequences (e.g., paperfolding words), introduce new operations that extend the space of possible queries, and show how to process the results. We begin with the fundamental concepts and problems related to automatic sequences, and the corresponding numeration systems. Building on that foundation, we discuss the general logical framework that formalizes the questions we can mechanically answer. We start with a first-order logical theory, and then extend it with additional predicates and operations. Then we explain a slightly different technique that works on a monadic second- order theory, but show that it is ultimately subsumed by an extension of the first-order theory. Next, we give two applications: critical exponent and paperfolding words. In the critical exponent example, we mechanically construct an automaton that describes a set of rational numbers related to a given automatic sequence. Then we give a polynomial-time algorithm to compute the supremum of this rational set, allowing us to compute the critical exponent and many similar quantities. In the paperfolding example, we extend our mechanical procedure to the paperfolding words, an uncountably infinite collection of infinite words. In the following chapter, we address abelian and additive problems on automatic sequences. We give an example of a natural predicate which is provably inexpressible in our first-order theory, and discuss alternate methods for solving abelian and additive problems on automatic sequences. We close with a chapter of open problems, drawn from the earlier chapters.

In recent years quantum algorithms have become a popular area of mathematical research. Farhi and Gutmann introduced the concept of a quantum walk in 1998. In this thesis we investigate mixing properties of continuous-time quantum walks from a mathematical perspective. We focus on the connections between mixing properties and association schemes. There are three main goals of this thesis. Our primary goal is to develop the algebraic groundwork necessary to systematically study mixing properties of continuous-time quantum walks on regular graphs. Using these tools we achieve two additional goals: we construct new families of graphs that admit uniform mixing, and we prove that other families of graphs never admit uniform mixing. We begin by introducing association schemes and continuous-time quantum walks. Within this framework we develop specific algebraic machinery to tackle the uniform mixing problem. Our main algebraic result shows that if a graph has an irrational eigenvalue, then its transition matrix has at least one transcendental coordinate at all nonzero times. Next we study algebraic varieties related to uniform mixing to determine information about the coordinates of the corresponding transition matrices. Combining this with our main algebraic result we prove that uniform mixing does not occur on even cycles or prime cycles. However, we show that the probability distribution of a quantum walk on a prime cycle gets arbitrarily close to uniform. Finally we consider uniform mixing on Cayley graphs of elementary abelian groups. We utilize graph quotients to connect the mixing properties of these graphs to Hamming graphs. This enables us to find new results about uniform mixing on Cayley graphs of certain elementary abelian groups.

Reed's conjecture is a proposed upper bound for the chromatic number of a graph. Reed's conjecture has already been proven for several families of graphs. In this paper, I show how one of those families of graphs can be extended to include additional graphs and also show that Reed's conjecture holds for a family of graphs known as cycle-power graphs, and also for their complements.

<p>This dissertation refines and extends the theory of mesoprimary decomposition, as introduced by Kahle and Miller. We begin with an overview of the existing theory of mesoprimary decomposition </p><p>in both the combinatorial setting of monoid congruences and the arithmetic setting of binomial ideals. We state all definitions and results that are relevant for subsequent chapters. </p><p>We classify redundant mesoprimary components in both the combinatorial and arithmetic settings. Kahle and Miller give a class of redundant components in each setting that are redundant in every mesoprimary decomposition. After identifying a further class of redundant components at the level of congruences, we give a condition on the associated monoid primes that guarantees the existence of unique irredundant mesoprimary decompositions in both settings. </p><p>We introduce soccular congruences as combinatorial approximations of irreducible binomial quotients and use the theory of mesoprimary decomposition to give a combinatorial method of constructing irreducible decompositions of binomial ideals. We also demonstrate a binomial ideal which does not admit a binomial irreducible decomposition, answering a long-standing problem of Eisenbud and Sturmfels. </p><p>We extend mesoprimary decomposition of monoid congruences to congruences on monoid modules. Much of the theory for monoid congruences extends to this new setting, including a characterization of mesoprimary monoid module congruences in terms of associated prime monoid congruences and a method for constructing coprincipal decompositions of monoid module congruences using key witnesses. </p><p>We conclude with a collection of open problems for future study.</p> / Dissertation