Combinatorics and Optimization

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This is the collection for the University of Waterloo's Department of Combinatorics and Optimization.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Now showing 1 - 20 of 459

Combinatorial Aspects of Feynman Integrals and Causal Set Theory
(University of Waterloo, 2025-05-23) Shaban, Kimia
This thesis consists of two distinct sections, the first highlighting causal set theory (CST) and the second focusing on Feynman period estimation. This work specifically covers how discrete structures from algebraic combinatorics can be applied to problems from physics, and how we can use computational tools and techniques to help solve these problems from a mathematical perspective. In the first part of the thesis, we introduce CST, an approach to quantum gravity that focuses on how events in spacetime are causally related to one another. In CST, discretized spacetime is given by a locally finite poset. A significant portion of this section focuses on covtree, which allows us to evolve such a discrete spacetime, in a way that does not depend on arbitrary labelling. However recognizing nodes of covtree involve solving a particular downset reconstruction problem. To attack this we define a graph that compares downsets that differ by exactly one element, with a particular focus on the order dimension two case. This section of the thesis sheds light on evolving a spacetime by constructing future elements within the covtree framework. Reconstructing spacetime in this way will allow for researchers to advance our understanding of covtree’s structure and improves causal set theory as an approach to quantum gravity. In the second part, we provide an introduction to Feynman periods, explaining their significance in quantum field theory (QFT) calculations. We will discuss how machine learning models, such as linear and quadratic regression, and graph neural networks, can be applied to Feynman graphs and their properties, to predict the Feynman period. Even simple techniques like linear regression, on graph parameters only, which do not consider the graph structure directly, can make highly accurate predictions of Feynman periods. The work presented in this section, done jointly with Dr. Paul-Hermann Balduf, is published in the Journal of High Energy Physics. This research has significant implications for QFT computations, which can become computationally infeasible at large scales, and facilitates further exploration in particle physics.
Analysis of the Three-operator Davis-Yin Splitting in the Inconsistent Case
(University of Waterloo, 2025-05-20) Naguib, Andrew
This thesis analyzes the Davis–Yin three-operator splitting method in the inconsistent case, where the underlying monotone inclusion problem may fail to have a solution. The Davis–Yin algorithm extends the Douglas–Rachford and forward–backward splitting methods and is effective in reformulating optimization and inclusion problems as fixed-point iterations. Our study investigates its behavior when no fixed point exists. We prove, under mild assumptions, that the Davis–Yin shadow sequence converges to a solution of the normal problem, which represents a minimal perturbation of the original formulation.
Quantum programming and synthesis: Internalizing Clifford operations and beyond
(University of Waterloo, 2025-04-29) Winnick, Samuel
Clifford operations are a subset of quantum operations used extensively in quantum error correction and classical simulation of quantum circuits. The first part of the thesis is motivated by the problem of programming with generalized Clifford operations, such as the quantum Fourier transform. We delve into the algebraic complications that arise for systems of even dimension $d$, and we are particularly interested in the case when $d$ is a power of $2$. We apply our results in the design of a quantum functional programming language, in which the user does not have to worry about these irrelevant complications. Later, we consider the problem of compiling circuits over universal gate sets. In particular, we study the problem of multi-qutrit exact synthesis over a variety of gate sets including Clifford gates. Lastly, we present a framework for defining a symplectic form on an object in a sufficiently structured category, and lay out the theory, generalizing the theory of symplectic forms on a finite dimensional vector space or locally compact abelian group. In the process, we develop new results and perspectives on operations defined on categories.
Asymptotics of the number of lattice points in the transportation polytope via optimization on Lorentzian polynomials.
(University of Waterloo, 2025-04-28) Lee, Thomas
We formally extend the theory of polynomial capacity to power series and totally uni- modular matrices. Using these results, we prove the log-asymptotic correctness of bounds by Brändén, Leake, and Pak developed through the use of Lorentzian polynomials ([BLP23]) under certain conditions, and provide a counterexample where these bounds are not log- asymptotically correct, even when symmetry exists.
Some Sylvester-Gallai-Type Theorems for Higher-Dimensional Flats
(University of Waterloo, 2025-04-28) Kroeker, Matthew Eliot
In response to an old question of Sylvester, Gallai proved that, for any finite set of points in the plane, not all contained in a single line, there is a line containing exactly two of them. This is known as the Sylvester-Gallai Theorem. In the language of matroid theory, which is used in this thesis, the Sylvester-Gallai Theorem says that every rank-3 real-representable matroid has a two-point line. Independently of Gallai and around the same time, Melchior proved the stronger result that, in a rank-3 real-representable matroid, the average number of points in a line is less than three. The purpose of this thesis is to prove theorems of this type for flats of higher dimension. The Sylvester-Gallai Theorem was generalized to higher dimensions by Hansen, who proved that, for any positive integer k, a simple real-representable matroid of rank at least 2k-1 has a rank-k independent flat. We will prove a theorem of this type for matroids representable over any given finite field by characterizing the "unavoidable flats" for matroids in this class. In particular, we show that any simple matroid of sufficiently high rank representable over a given finite field has a rank-k flat which is either independent or a projective or affine geometry. As a corollary, we will get the following Ramsey theorem: for any two-colouring of the points of a matroid of sufficiently high rank representable over a given finite field, there is a monochromatic rank-k flat. The rest of the thesis concerns generalizing Melchior’s theorem to higher dimensions. For planes, we prove that, in a real-representable matroid M of rank at least four, the average plane-size is at most an absolute constant, unless M is a direct sum of lines (in which case the average plane-size could be arbitrarily large). Generalizing this result to hyperplanes of arbitrary rank k, we prove that, if M is a simple rank-(k+1) real-representable matroid whose average hyperplane-size is greater than some constant depending only on k, then all but a constant number b_k of the points of M are covered by a small collection of flats of relatively low rank. We also show that this constant b_k is best-possible. Melchior’s Theorem also implies that the average size of a flat (of any rank) in a rank-3 real-representable matroid is at most an absolute constant. We prove that, for any positive integer r, the average size of a flat in a rank-r real-representable matroid is at most some constant depending only on r.
Denormalized Lorentzian Laurent series
(University of Waterloo, 2025-04-24) Mohammadi Yekta, Maryam
We introduce and study the class of denormalized Lorentzian Laurent series and prove lower bounds on their coefficients. To do this, we define a capacity function on Laurent series using their domain of convergences. As a result, we can use simpler arguments to re-prove an already known lower bound on the number of contingency tables, which are lattice points of transportation polytopes; and tidy up and slightly improve an already known lower bound on the number of integral flows of a network.
Max-Min Greedy Matching
(University of Waterloo, 2024-12-17) Rakhmatullin, Ramazan; Pashkovich, Kanstantsin
We study the problem of max-min greedy matching introduced in (Eden et al. 2022). The max-min greedy matching problem is a variant of the well known online bipartite matching problem introduced in (Karp et al. 1990). Given a bipartite graph $G=(U \cup V; E)$, the first player selects a priority ordering $\pi$ of $V$. After that the second player selects an arrival order $\sigma$ of $U$ depending on the ordering $\pi$ selected by the first player. Once both players did their selections, the vertices of $U$ arrive in the order specified by $\sigma$. Upon arrival, each vertex in $U$ is matched with its highest-priority neighbor that is still unmatched, if one exists. Here, the highest-priority is defined with respect to $\pi$. The first player aims to maximize the size of the resulting matching, while the second player seeks to minimize it. We investigate the max-min greedy matching problem. We aim to improve the lower bound of the guaranteed matching size achievable by the first player. We present a linear time algorithm that constructs an ordering $\pi$ such that for every $\sigma$ the ratio between the resulting matching size and the maximum matching size is at least $\frac{5}{9}$. Our result improves the previous best known ratio of $\frac{1}{2} + \frac{1}{86}$ from (Eden et al. 2022). The best known upper bound on the ratio is $\frac{2}{3}$ (Cohen-Addad et al. 2016). Thus, our result makes the gap between the lower and upper bounds for the ratio substantially smaller. Our method introduces a novel ``balanced path decomposition'' technique. This decomposition provides crucial structural properties such as existence of large matchings within these paths; and absence of certain edges between the paths. These structural properties allow us to prove the desired lower bound on the ratio. Our findings suggest that the achieved matching ratio can be further improved and that the balanced path decomposition may have broader applications.
On the relation among the maximum degree, the chromatic number and the clique number
(University of Waterloo, 2024-12-12) Xie, Xinyue; Haxell, Penny
We study the relation among the maximum degree $\Delta(G)$, the chromatic number $\chi(G)$ and the clique number $\omega(G)$ of a graph $G$ by proving results related to the Borodin-Kostochka Conjecture and Reed's Conjecture, which are two long-standing conjectures on this subject. The Borodin-Kostochka Conjecture states that a graph $G$ with $\chi(G)=\Delta(G)\ge9$ has a clique of size $\Delta(G)$. Reed's Conjecture states that any graph $G$ satisfies $\chi(G)\le \lceil\frac{1}{2}(\Delta(G)+\omega(G)+1)\rceil$. By combining and extending the approaches taken by Cranston and Rabern and by Haxell and MacDonald, we use the technique of Mozhan partitions to prove a generalized Borodin-Kostochka-type result as follows. Given a nonnegative integer $t$, for every graph $G$ with $\Delta(G)\ge 4t^2+11t+7$ and $\chi(G)=\Delta(G)-t$, the graph $G$ contains a clique of size $\Delta(G)-2t^2-7t-4$. We further prove that both conjectures hold for graphs with at least one critical vertex that is not in an odd hole, by following similar proofs as by Aravind, Karthick and Subramanian and by Chen, Lan, Lin and Zhou. This result then motivates us to study Reed's Conjecture in the context of graphs in which all vertices are in some odd hole. By investigating blow-ups and powers of cycles, we enumerate several classes of graphs for which Reed's Conjecture holds with equality. They generalize many of the known tight examples. We also introduce a class of graphs constructed from blow-ups and powers of paths, which gives rise to a new family of irregular tight examples.
Algebraic Approach to Quantum Isomorphisms
(University of Waterloo, 2024-09-24) Sobchuk, Mariia; Godsil, Chris
In very brief, this thesis is a study of quantum isomorphisms. We have started with two pairs of quantum isomorphic graphs and looked for generalizations of those. We have learned that those two pairs of graphs are related by Godsil-McKay switching and one of the graphs is an orthogonality graph of lines in a root system. These two observations lead to research in two directions. First, since quantum isomorphisms preserve coherent algebras, we studied a question of when Godsil-McKay switching preserved coherent algebras. In this way, non isomorphic graphs related by Godsil-McKay switching with isomorphic coherent algebras are candidates to being quantum isomorphic and non isomorphic. Second, while it was known that one of the graphs in a pair was an orthogonality graphs of the lines in a root system $E_8,$ we showed that a graph from another pair is also an orthogonality graph of the lines in a root system $F_4.$ We have studied orthogonality graphs of lines in root systems $B_{2^d},C_{2^d},D_{2^d}$ and showed that they have quantum symmetry. Finally, we have touched upon structures of quantum permutations, relationships between fractional and quantum isomorphisms as well as connection to quantum independence and chromatic numbers.
Theory and Results on Restarting Schemes for Accelerated First Order Methods
(University of Waterloo, 2024-09-10) Pavlovic, Viktor; Vavasis, Stephen; Moursi, Walaa
Composite convex optimization problems are abundant in industry, and first order methods to solve them are growing in popularity as the size of variables reaches billions. Since the objective function could be possibly non-smooth, proximal gradient methods are one of the main tools for these problems. These methods benefit from acceleration, which uses the memory of past iterates to add momentum to the algorithms. Such methods have a O(1/k^2) convergence rate in terms of function value where k is the iteration number. Restarting algorithms has been seen to help speed up algorithms. O'Donoghue and Candes introduced adaptive restart strategies for accelerated first order methods which rely on easy to compute conditions, and indicate a large performance boost in terms of convergence. The restart works by resetting the momentum gained from acceleration. Their strategies in general are a heuristic, and there is no proof of convergence. In this thesis we show that restarting with the O'Donoghue and Candes condition improves the standard convergence rate in special cases. We consider the case of one-dimensional functions where we prove that the gradient based restart strategy from O'Donoghue and Candes improves the O(1/k^2) bound. We also study the restarting scheme applied to the method of alternating projections (MAP) for two closed, convex, and nonempty sets. It is shown in Chapter 6 that MAP falls into the convex composite paradigm and therefore acceleration can be applied. We study the case of MAP applied to two hyperplanes in arbitrary dimension. Furthermore we make observations as to why the restarts help, what makes a good restart condition, as well as what is needed to make progress in the general case.
Tubings of Rooted Trees
(University of Waterloo, 2024-09-05) Cantwell, Amelia; Yeats, Karen
A tubing of a rooted tree is a broad term for a way to split up the tree into induced connected subtrees. They are useful for computing series expansion coefficients. This thesis discusses two different definitions of tubings, one which helps us understand Dyson- Schwinger equations, and the other which helps us understand the Magnus expansion. Chord diagrams are combinatorial objects that relate points on a circle. We can explicitly map rooted connected chord diagrams to tubings of rooted trees by a bijection, and we explore further combinatorial properties arising from this map. Furthermore, this thesis describes how re-rooting a tubed tree will change the chord diagram. We present an algorithm for finding the new chord diagram by switching some chords around. Finally, a different notion of tubings of rooted trees is introduced, which was originally developed by Mencattini and Quesney [27]. They defined two sub-types of tubings: vertical and horizontal which are used to find coefficients in the Magnus expansion. These two types of tubings have an interesting relationship when the forests are viewed as plane posets.
Tight Multi-Target Security for Key Encapsulation Mechanisms
(University of Waterloo, 2024-09-04) Glabush, Lewis; Stebila, Douglas
The use of symmetric encryption schemes requires that the communicating parties have access to a shared secret key. A key encapsulation mechanism (KEM) is a cryptographic tool for the secure establishment of such a key. The KEMs most commonly used at this time are vulnerable to adversaries with access to a large quantum computer. This project concerns KEMs that are resistant to all known quantum attacks, such as lattice-based schemes. A desirable property for any KEM is multi-target security, capturing the idea that security does not degrade below the targeted level as the number of users of a protocol or the amount of communication per user scales to a certain threshold. For schemes based on prime-order groups, multi-ciphertext security can be trivially reduced to singleciphertext security using self reducibility arguments, but these arguments are not available for lattice-based schemes. Indeed, one of the alternates in NIST’s post-quantum cryptography standardization project, FrodoKEM, was susceptible to simple attacks in the multi-target setting. The standard approach to building IND-CCA secure KEMs has been to start with an IND-CPA secure public key encryption scheme (PKE) and apply the Fujisaki-Okamoto transform (FO). In this paper, we introduce a new variant of the FO transform, called the salted FO transform (SFO) which adds a uniformly random salt to the generation of ciphertexts. We then show that the resulting KEM’s have much tighter security bounds compared to their generic counterparts. We then apply our results to FrodoKEM to resolve the aforementioned simple attacks.
Price-setting Problems and Matroid Bayesian Online Selection
(University of Waterloo, 2024-08-30) DeHaan, Ian; Pashkovich, Kanstantsin
We study a class of Bayesian online selection problems with matroid constraints. Consider a seller who has several items they wish to sell, with the set of sold items being subject to some structural constraints, e.g., the set of sold items should be independent with respect to some matroid. Each item has an offer value drawn independently from a known distribution. In a known order, items arrive and drawn values are presented to the seller. If the seller chooses to sell the item, they gain the drawn value as revenue. Given distribution information for each item, the seller wishes to maximize their expected revenue by carefully choosing which offers to accept as they arrive. Such problems have been studied extensively when the seller's revenue is compared with the offline optimum, referred to as the "prophet". In this setting, a tight 2-competitive algorithm is known when the seller is limited to selling independent sets of a matroid [KW12]. We turn our attention to the online optimum, or "philosopher", and ask how well the seller can do with polynomial-time computation, compared to a seller with unlimited computation but with the same limited distribution information about offers. We show that when the underlying constraints are laminar and the arrival of buyers follows a natural "left-to-right" order, there is a polynomial-time approximation scheme for maximizing the seller's revenue. We also show that such a result is impossible for the related case when the underlying constraints correspond to a graphic matroid. In particular, it is PSPACE-hard to approximate the philosopher's expected revenue to some fixed constant α < 1; moreover, this cannot be alleviated by requirements on the arrival order in the case of graphic matroids. We also show similar hardness results for both transversal and cographic matroids. We then turn our attention to a related problem where the arrival order of items is unknown and uniformly random. In this setting, we show that there is a polynomial-time approximation scheme whenever the rank of the matroid is bounded above by a constant and all probabilities in the input are bounded below by a constant. We additionally examine the computational complexity of computing the prophet's expected revenue. We show that this problem is #P-hard, and give a fully polynomial-time randomized approximation scheme for the problem.
Contracts for Density and Packing Functions
(University of Waterloo, 2024-08-30) Skitsko, Jacob; Pashkovich, Kanstantsin
We study contracts for combinatorial problems in multi-agent settings. In this problem, a principal designs a contract with several agents, whose actions the principal is unable to observe. The principal is able to see only the outcome of the agents' collective actions. All agents that decided to exert effort incur costs, and so naturally all agents expect a fraction of the principal's reward as a compensation. The principal needs to decide what fraction of their reward to give to each agent so that the principal's expected utility is maximized. One of our focuses is on the case when the principal's reward function is supermodular and is based on some graph. Recently, Deo-Campo Vuong et al. showed that for this problem it is impossible to provide any finite multiplicative approximation or additive FPTAS unless P=NP. On a positive note, Deo-Campo Vuong et al. provided an additive PTAS for the case when all agents have the same cost. Deo-Campo Vuong et al. asked whether an additive PTAS can be obtained for the general case, i.e for the case when agents potentially have different costs. In this thesis, we answer this open question in positive. Additionally, we provide multiplicative approximation algorithms for functions that are based on hypergraphs and encode packing constraints. This family of functions provides a generalization for XOS functions.
Control over the KKR bijection with respect to the nesting structure on rigged configurations and a CSP instance involving Motzkin numbers
(University of Waterloo, 2024-08-27) Chan, William; Mandelshtam, Olya
There are two disjoint main projects that this thesis covers. The Motzkin numbers are a sort of “re- laxed version” of the Catalan numbers. For example, Catalan numbers count perfect non-crossing matchings, while Motzkin numbers count not necessarily perfect non-crossing matchings. The first project deals with instances of the cyclic sieving phenomenon involving Motzkin numbers and their standard q−analogue. We also show that the standard q−analogue for Motzkin numbers satisfy a similar generating series interpretation to that of the q−Catalan numbers. The second project deals with understanding the Kerov-Kirillov-Reshetikhin (KKR) bijection between semistandard tableaux and rigged configurations with a particular emphasis on the standard case. In partic- ular, we understand inducing perturbations on the corresponding rigged configuration via direct operations on the tableau. We develop a technique to take a standard tableau T, and output a new standard tableau T′ which has the same corresponding rigged configuration up to a rigging on any desired row of the first rigged partition. We also develop an alternate technique to Kuniba, Okado, Sakamoto, Takagi, and Yamada that “unwraps” the natural nesting structure on rigged configurations. The primary operation from which all the above follows from is “raise” which we introduce and give various combinatorial models for. The operation raise induces a very simple, controlled perturbation on the corresponding rigged configuration. Our results are formulated in terms of paths or (classically) highest weight elements of tensor products of Kirillov-Reshetikhin crystals.
Euclidean Distance Matrix Correction with a Single Corrupted Element
(University of Waterloo, 2024-08-27) Tina Chenrui, Xu; Wolkowicz, Henry; Walaa, Moursi
In this thesis, we study the problem of correcting the error from a noisy Euclidean distance matrix (EDM). An EDM is a matrix where elements are squared distance between points in R^d. We consider the special case where only one of the distance is corrupted. We develop efficient algorithms to solve this problem, initially assuming that the points are in general position, and solve the problem using three different types of facial reduction: exposing vector, facial vectors, and Gale transform. Furthermore, we investigate yielding elements of an EDM and develop an algorithm for identifying one small principal submatrix with the embedding dimension d when many of the points are in a linear manifold of dimension smaller than d, allowing us to handle a more general problem. We present numerical experiments implemented in MATLAB, demonstrating the effectiveness of our solutions.
Non-Generic Analytic Combinatorics in Several Variables and Lattice Path Asymptotics
(University of Waterloo, 2024-08-27) Kroitor, Alexander; Melczer, Stephen
Finding and analyzing generating functions is a classic and powerful way of studying combinatorial structures. While generating functions are a priori purely formal objects, there is a rich theory treating them as complex-analytic functions. This is the field of analytic combinatorics, which exploits classical results in complex and harmonic analysis to find asymptotics of coefficient sequences of generating functions. More recently, the field of analytic combinatorics in several variables has been developed to study multivariate generating functions, which are amenable to multivariate analytic results. When dealing with multivariate asymptotics we first pick a direction vector. To find asymptotics using analytic combinatorics, one typically uses the Cauchy integral formula to write the underlying sequence being studied as a complex integral, then uses residue computations to reduce to a so-called Fourier-Laplace integral. When the direction vector chosen is generic, the Fourier-Laplace integral being studied is well-behaved and asymptotics can be computed. When the direction vector chosen is non-generic, the Fourier-Laplace integral has singular amplitude, and asymptotics are harder to compute. This thesis studies singular Fourier-Laplace integrals and their applications to combinatorics. Determining asymptotics for the number of lattice walks restricted to certain regions is one of the particular successes of analytic combinatorics in several variables. In particular, Melczer and Mishna determined asymptotics for the number of walks in an orthant whose set of steps is a subset of { ±1, 0 }^d \ { 0 } and is symmetric over every axis. Melczer and Wilson later determined asymptotics for lattice path models whose set of steps is a subset of { ±1, 0 }^d \ { 0 } and is symmetric over all but one axis, except when the vector sum of all the steps is 0. Here we determine asymptotics in the zero sum case using asymptotics of singular Fourier-Laplace integrals. We additionally study asymptotics of lattice path models restricted to Weyl chambers, before giving some generalizations of asymptotics of singular Fourier-Laplace integrals.
Fault-tolerant Preparation of Distant Logical Bell Pair - with application in the Magic Square Game
(University of Waterloo, 2024-08-23) Liu, Andy Zeyi; Leung, Debbie
Quantum entanglement facilitates nonlocal correlations that defy classical physics, forming the basis for quantum technologies, including quantum computation and communication. Nonlocal games exemplify this power, where entangled players achieve outcomes unattainable by classical means. This thesis focuses on optimizing the preparation of high-fidelity logical Bell pairs in the context of the fault-tolerant magic square game, seeking to minimize Bell pair(ebit) consumption while maintaining a low logical error rate. In this thesis, we introduce a novel approach leveraging an interface circuit and entanglement purification protocol (EPP) to translate states between physical and logical qubits and purify noisy logical ebits. This method significantly reduces the number of initial ebits needed compared to conventional strategies. Our analytical and numerical analyses, particularly for the [[7k,1,3k]] concatenated Steane code, demonstrate substantial (actually, exponential) ebit savings and higher noise threshold. Analytical lower bounds for local noise threshold of 4.70 × 10−4 and initial ebit infidelity threshold of 18.3% are obtained. Additionally, we construct an analogous interface for the surface code through lattice surgery, offering further improvements in fault tolerance and compatibility with current quantum hardware. Our framework is adaptable to various quantum error-correcting codes (QECCs) and experimental platforms. We hope our protocol will not only enhance understanding of fault-tolerant nonlocal games, but also spark further exploration of interfacing between different QECCs, promoting the development of modular quantum architectures and advancing quantum internet.
Degrees of P -Grothendieck polynomials and regularity of Pfaffian varieties
(University of Waterloo, 2024-08-23) St.Denis, Matthew; Pechenik, Oliver
We prove a formula for the degrees of Ikeda and Naruse’s P -Grothendieck polynomials using combinatorics of shifted tableaux. We show this formula can be used in conjunction with results of Hamaker, Marberg, and Pawlowski to obtain an upper bound on the Castelnuovo–Mumford regularity of certain Pfaffian varieties known as vexillary skew-symmetric matrix Schubert varieties. Similar combinatorics additionally yields a new formula for the degree of Grassmannian Grothendieck polynomials and the regularity of Grassmannian matrix Schubert varieties, complementing a 2021 formula of Rajchgot, Ren, Robichaux, St. Dizier, and Weigandt.
Arboricity and transversal problems on bounded degree graphs
(University of Waterloo, 2024-08-19) Wdowinski, Ronen; Haxell, Penny
In this thesis, we investigate arboricity and transversal problems on graphs in a bounded degree setting. We pay particular attention to problems involving both the maximum degree and maximum density of the graphs. The first topic is bounded degree arboricity. The arboricity of a multigraph $G$ is the minimum number of forests required to cover its edge set. It is well understood through a theorem of Nash-Williams. For the problem of bounded degree arboricity, we seek to cover the edge set of a multigraph $G$ by the minimum number of forests such that every vertex $v$ has maximum degree at most $f(v)$ in every forest, for some fixed weight function $f : V(G) \rightarrow \mathbb{Z}_{\ge 2}$. The case $f = 2$ is referred to as linear arboricity. The Linear Arboricity Conjecture, due to Akiyama, Exoo, and Harary, asserts that the linear arboricity of a simple graph $G$ with maximum degree $\Delta$ is at most $\lceil (\Delta+1)/2 \rceil$. Using tools from orientations of multigraphs, we prove that the Linear Arboricity Conjecture holds for multigraphs whose maximum degree is sufficiently large compared to a certain maximum density. This improves on previous results of this type, and moreover our methods extend to give a general upper bound on the bounded degree arboricity of multigraphs, as a maximum of a weighted maximum degree and maximum density of the multigraph. In addition, we disprove a conjecture of Truszczy\'nski which proposes a more precise upper bound on the bounded degree arboricity of multigraphs, but we also show that the conjecture does hold for simple graphs with sufficiently large girth, and that it holds for all simple graphs asymptotically. The second topic in this thesis is independent transversals in bounded degree graphs. Given a graph $G$ and a partition $\mathcal{P} = \{V_1, \ldots, V_n\}$ of its vertex set, an independent transversal is a transversal of $\mathcal{P}$ that is independent in $G$. Haxell proved that there exists an independent transversal when $G$ has maximum degree $\Delta$ and $|V_i| \ge 2\Delta$ for every $i$, and this condition has been shown to be best possible by Szab\'o and Tardos. Wanless and Wood proved the existence of an independent transversal when the maximum class average degree (a kind of maximum density) of $G, \mathcal{P}$ is $D$ and $|V_i| \ge 4D$ for every $i$, and this condition has been shown to be asymptotically best possible by Groenland, Kaiser, Treffers, and Wales. Using topological methods, we find an interpolation between these two sufficient conditions, and moreover we give a construction showing that our result is best possible. The method for our tight construction, based on a simple lemma, is then further explored. We use our method to streamline tight constructions for Haxell's theorem, and we prove that our method in fact yields all possible extremal constructions. We also use it to give more streamlined counterexamples to a list coloring conjecture of Reed. Then we adapt the construction method to the setting of full rainbow matchings of multi-hypergraphs, which are independent transversals in their line graphs. Aharoni, Berger, and Meshulam proved that there exists a full rainbow matching in an $r$-uniform multigraph $G$ with maximum degree $\Delta$ when every edge class has size $|E_i| \ge r\Delta$. We use our method to give tight constructions for this theorem. We also describe counterexamples to a color degree generalization of Galvin's Theorem on list edge-coloring bipartite multigraphs. Finally, we describe how our method applies to independent transversals in uniform hypergraphs and other related settings.

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