Combinatorics and Optimization

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

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Some Applications of Combinatorial Hopf Algebras to Integro-Differential Equations and Symmetric Function Identities
(University of Waterloo, 2024-07-09) Olson-Harris, Nicholas
Hopf algebras built from combinatorial objects have found application both within combinatorics and, following the work of Connes and Kreimer, in quantum field theory. Despite the apparent gulf between these areas, the types of Hopf algebras that arise are very similar. We use Hopf algebra techniques to solve two problems, one coming from quantum field theory and one from algebraic combinatorics. (1) Dyson–Schwinger equations are a formulation of the equations of motion of quantum field theory. From a mathematical perspective they are integro-differential equations which have a recursive, tree-like structure. In some cases, these equations are known to have solutions which can be written as combinatorial expansions over connected chord diagrams. We give a new expansion in terms of rooted trees equipped with a kind of decomposition we call a binary tubing. This is similar to the chord diagram expansion, but holds in greater generality, including to systems of Dyson–Schwinger equations and to Dyson–Schwinger equations in which insertion places are distinguished by different variables in the Mellin transform. Moreover we prove these results as a direct application of a purely Hopf-algebraic theorem characterizing maps from the Connes–Kreimer Hopf algebra of rooted trees (and variants thereof) to the Hopf algebra of univariate polynomials which arise from the universal property of the former. (2) A pair of skew Ferrers shapes are said to be skew-equivalent if they admit the same number of semistandard Young tableaux of each weight, or in other words if the skew Schur functions they define are equal. McNamara and van Willigenburg conjectured necessary and sufficient combinatorial conditions for this to happen but were unable to prove either direction in complete generality. Using Hopf-algebraic techniques building on a partial result of Yeats, we prove sufficiency.
Chromatic Number of Random Signed Graphs
(University of Waterloo, 2024-05-03) Yuan, Dao Chen
We naturally extend Bollobas's classical method and result about the chromatic number of random graphs chi(G(n,p)) ~ n/log_b(n) (for p constant, b=1/(1-p)) to the chromatic number of random signed graphs to obtain chi(G(n,p,q)) ~ n/log_b(n) (for p constant, b=1/(1-p), q=o(1)). We also give a sufficient bound on q under which a.a.s. the chromatic number of G(n,p,q) is unchanged before and after adding negative edges.
Routing, Scheduling, and Sorting in Consolidated Networks
(University of Waterloo, 2024-04-25) Van Dyk, Madison
Modern parcel logistic networks are designed to ship demand between given origin, destination pairs of nodes in an underlying directed network. Efficiency dictates that volume needs to be consolidated at intermediate nodes in typical hub-and-spoke fashion. In practice, such consolidation requires tracking packages in both space and time (temporal network design), as well as parcel sortation. In the first half of the thesis, we study solution methods for temporal problems arising in consolidated networks. While many time-dependent network design problems can be formulated as time-indexed formulations, the size of these formulations depends on the discretization of the time horizon and can become prohibitively large. The recently-developed dynamic discretization discovery (DDD) method allows many time-dependent problems to become more tractable by iteratively solving instances of the problem on smaller networks where each node has its own discrete set of departure times. There are two design elements of existing DDD algorithms that make it problematic for use in region-based hub-and-spoke networks. First, in each iteration, all arcs departing a common node share the same set of departure times. This causes DDD to be ineffective for solving problems where all near-optimal solutions require many distinct departure times at the majority of the high-degree nodes in the network, an aspect characteristic of region-based networks. A second challenge is handling static storage constraints without leading to a weak relaxation in each iteration. To mitigate these limitations, an arc-based DDD framework is proposed in Chapter 3, where departure times are determined at the arc level instead of the node level. We apply this arc-based DDD method to instances of the service network design problem (SND). We show that an arc-based approach is particularly advantageous when instances arise from region-based networks, and when candidate paths are fixed in the base graph for each commodity. Moreover, our algorithm builds upon the existing DDD framework and achieves these improvements with only benign modifications to the original implementation. Additionally, Chapter 4 introduces bounds on additional storage required in each iteration, expanding the applicability of DDD to problems with bounded node storage, such as the universal packet routing problem. Our arguments rely solely on the structure of the standard map, μ, from the original formulation to the smaller relaxed formulations. In order to maintain consistent operations, some models stipulate that the implemented transportation schedule must be repeated each day. In Chapter 5 we present a DDD model for solving a version of SND with cyclic constraints. We show that these cyclic constraints require new conditions on the time discretization at each node, leading to larger partial networks. We then highlight challenges in reducing the size of partial networks as they grow over each iteration of DDD. We demonstrate that certain policies for removing departure times in each iteration can cause the iterations in DDD to repeat, preventing termination. In the second half of this thesis, we study parcel sortation, an aspect of routing that has previously been left unaddressed from a theory perspective. Warehouses have limited sort points, the physical devices tasked with handling packages destined for a particular downstream warehouse. We propose a mathematical model for the physical requirements, and limitations of parcel sortation. We then show that it is NP-hard to determine whether a feasible sortation plan exists. We consider two natural objectives: minimizing the maximum number of sort points required at a warehouse, and minimizing the total number of sort points required in the network. In Chapter 6, we consider the problem of minimizing the maximum number of sort points required at a warehouse. We discuss several settings, where (near-)optimality of a given sortation instance can be determined efficiently. The algorithms we propose are fast and build on combinatorial witness set type lower bounds that are reminiscent and extend those used in earlier work on degree-bounded spanning trees and arborescences. In Chapter 7, we present algorithms for minimizing the total number of sort points required in a network. In contrast to the min-degree setting, it is not known if this min-cardinality setting is NP-hard. In progress towards answering this question, we present fast combinatorial algorithms for solving in-tree, out-tree, and spider instances. Our algorithms are based on reduction, decomposition, and uncrossing techniques that simplify instances.
Analytic Methods and Combinatorial Plants
(University of Waterloo, 2024-04-08) Chizewer, Jeremy
Combinatorial structures have broad applications in computer science, from error-correcting codes to matrix multiplication. Many analytic tools have been developed for studying these structures. In this thesis, we examine three applications of these tools to problems in combinatorics. By coincidence, each problem involves a combinatorial structure named for a plant--AVL trees, cactus graphs, and sunflowers--which we refer to collectively as combinatorial plants. In our first result, we use a novel decomposition to create a succinct encoding for tree classes satisfying certain properties, extending results of Munro, Nicholson, Benkner, and Wild. This has applications to the study of data structures in computer science, and our encoding supports a wide range of operations in constant time. To analyze our encoding, we derive asymptotics for the information-theoretic lower bound on the number of bits needed to store these trees. Our method characterizes the exponential growth for the counting sequence of combinatorial classes whose generating functions satisfy certain functional equations, and may be of independent interest. Our analysis applies to AVL trees (a commonly studied self-balancing binary search tree in computer science) as a special case, and we show that about $0.938$ bits per node are necessary and sufficient to encode AVL trees. Next, we study the hat guessing game on graphs. In this game, a player is placed on each vertex $v$ of a graph $G$ and assigned a colored hat from $h(v)$ possible colors. Each player makes a deterministic guess on their hat color based on the colors assigned to the players on neighboring vertices, and the players win if at least one player correctly guesses his assigned color. If there exists a strategy that ensures at least one player guesses correctly for every possible assignment of colors, the game defined by $\langle G,h\rangle$ is called winning. The hat guessing number of $G$ is the largest integer $q$ so that if $h(v)=q$ for all $v\in G$ then $\langle G,h\rangle$ is winning. We determine whether $\langle G,h\rangle $ is winning for any $h$ whenever $G$ is a cycle, resolving a conjecture of Kokhas and Latyshev in the affirmative and extending it. We then use this result to determine the hat guessing number of every cactus graph, in which every pair of cycles shares at most one vertex. Finally, we study the sunflower problem. A sunflower with $r$ petals is a collection of $r$ sets over a ground set $X$ such that every element in $X$ is in no set, every set, or exactly one set. Erd\H{o}s and Rado~\cite{er} showed that a family of sets of size $n$ contains a sunflower if there are more than $n!(r-1)^n$ sets in the family. Alweiss et al.~\cite{alwz}, and subsequently Rao~\cite{rao} and Bell et al.~\cite{bcw}, improved this bound to $(O(r \log(n))^n$. We study the case where the pairwise intersections of the set family are restricted. In particular, we improve the best-known bound for set families when the size of the pairwise intersections of any two sets is in a set $L$. We also present a new bound for the special case when the set $L$ is the nonnegative integers less than or equal to $d$, using techniques of Alweiss et al.~\cite{alwz}.
Graph-Theoretic Techniques for Optimizing NISQ Algorithms
(University of Waterloo, 2024-02-15) Jena, Andrew
Entering the NISQ era, the search for useful yet simple quantum algorithms is perhaps of more importance now than it may ever be in the future. In place of quantum walks, the quantum Fourier transform, and asymptotic results about far-term advantages of quantum computation, the real-world applications of today involve nitty-gritty details and optimizations which make the most use of our limited resources. These priorities pervade the research presented in this thesis, which focuses on combinatorial techniques for optimizing NISQ algorithms. With variational algorithms directing the discussion, we investigate methods for reducing Hamiltonians, reducing measurement errors, and reducing entangling gates. All three of these reductions bring us ever closer to demonstrating the utility of quantum devices, by improving some of the major bottlenecks of the NISQ era, and all of them do so while rarely ever leaving the combinatorial framework provided by stabilizer states. The qubit tapering approach to Hamiltonian simplification which we present was developed independently of the work by Bravyi et al., who discovered how to reduce qubit counts by parallelizing the computation of the ground state. The measurement scheme we describe, AEQuO, is built upon years of research and dozens of articles detailing, comparing, and contrasting a plethora of schemes. The circuit optimization technique we introduce answers a question posed by Adcock et al., and our ideas and proofs are fundamentally grounded in the literature of isotropic systems and the graph-based results which have followed from it.
Formalizing the Excluded Minor Characterization of Binary Matroids in the Lean Theorem Prover
(University of Waterloo, 2024-01-23) Gusakov, Alena
A matroid is a mathematical object that generalizes the notion of linear independence of a set of vectors to an abstract independence of sets, with applications to optimization, linear algebra, graph theory, and algebraic geometry. Matroid theorists are often concerned with representations of matroids over fields. Tutte's seminal theorem proven in 1958 characterizes matroids representable over GF(2) by noncontainment of U2,4 as a matroid minor. In this thesis, we document a formalization of the theorem and its proof in the Lean Theorem Prover, building on its community-built mathematics library, mathlib.
Implementing the Castryck-Decru attack on SIDH with general primes
(University of Waterloo, 2024-01-09) Laflamme, Jeanne
With the rapid progress of quantum computers in recent years, efforts have been made to standardize new public-key cryptographic protocols which would be secure against them. One of the schemes in contention was Supersingular Isogeny Diffie-Hellman (SIDH). This scheme relied on the assumed hardness of the isogeny problem on supersingular elliptic curves. However, in the SIDH protocol extra information on the secret isogenies is transmitted. In July 2022, Castryck and Decru found a way to exploit this information to completely break the scheme. They gave an implementation of their attack which allows to recover Bob’s secret key in a few seconds on a laptop. Usually, Alice and Bob’s secret isogenies are taken to have degree 2^a and 3^b respectively. This thesis gives a more general implementation of the attack in Magma which works even if Alice and Bob’s secret isogenies have degrees lA^a and lB^b for more general primes lA and lB.
Graphical CSS Code Transformation Using ZX Calculus
(University of Waterloo, 2023-12-21) Li, Sarah Meng
In this work, we present a generic approach to transform CSS codes by building upon their equivalence to phase-free ZX diagrams. Using the ZX calculus, we demonstrate diagrammatic transformations between encoding maps associated with different codes. As a motivating example, we give explicit transformations between the Steane code and the quantum Reed-Muller code, since by switching between these two codes, one can obtain a fault-tolerant universal gate set. To this end, we propose a bidirectional rewrite rule to find a (not necessarily transversal) physical implementation for any logical ZX diagram in any CSS code. We then focus on two code transformation techniques: code morphing, a procedure that transforms a code while retaining its fault-tolerant gates, and gauge fixing, where complimentary codes can be obtained from a common subsystem code (e.g., the Steane and the quantum Reed-Muller codes from the [[15, 1, 3, 3]] code). We provide explicit graphical derivations for these techniques and show how ZX and graphical encoder maps relate several equivalent perspectives on these code-transforming operations.
Combinatorially Thin Trees and Spectrally Thin Trees in Structured Graphs
(University of Waterloo, 2023-12-19) Alghasi, Mahtab
Given a graph $G=(V,E)$, finding simpler estimates of $G$ with possibly fewer edges or vertices while capturing some of its specific properties has been used in order to design efficient algorithms. The concept of estimating a graph with a simpler graph is known as graph sparsification. Spanning trees are an important family of graph sparsifiers that maintain connectivity of graphs, and have been utilized in many applications. However, spanning trees are a wide family, and for some applications one might need the spanning tree to have specific properties. Combinatorially thin trees are a type of spanning trees that show up in applications such as Asymmetric Travelling Salesman Problem (ATSP). A spanning tree $T$ of $G$ is combinatorially thin if there is no cut $U\subset V$ such that $T$ contains all the edges in $\delta(U)$, and the thinness parameter $\alpha_G(T)$ measures the maximum fraction of edges in $E(T)\cap \delta(U)$ compared to $\delta(U)$ over all cuts $U\subset V$. Intuitively, combinatorial thinness measures how much edge-connectivity we lose while removing the spanning tree $T$ from $G$. It is easy to verify that if $G$ has connectivity $k$, then $\frac{1}{k}$ lower bounds $\alpha_G$. On the other hand, Goddyn conjectured that $\alpha_G$ can also be upper bounded as a function of connectivity $\alpha_G = f(\frac{1}{k})$. This conjecture which is known as thin tree conjecture, was proved for the special case of graphs with bounded genus by Oveis-Gharan and Saberi, in 2011. However, the general case is still open. In the first part of this thesis, we study some of the known connections between edge-connectivity and $\alpha_{G}$ and investigate the result of Oveis-Gharan and Saberi for the special case of planar graphs. For a general graph $G$ and spanning tree $T$, even verifying the combinatorial thinness $\alpha_{G}(T)$ of $T$ is an $\text{NP}$-hard problem. A natural more efficiently computable relaxation of combinatorial thinness is the notion of spectral thinness. For a graph $G$ and a spanning tree $T$ in $G$ the spectral thinness $\theta_{G}(T)$ is the smallest value of $\theta$ such that $\theta\L_G - \L_T$ is a positive semidefinite matrix where $\L_G$ and $\L_T$ are Laplacian matrices of $G$ and $T$. Additionally, we define $\theta_G$ to be the minimum value of $\theta_{G}(T)$ over all spanning trees $T$ of $G$. Similar to combinatorial thinness and connectivity, $\theta_{G}(T)$ can be lower bounded by the maximum effective resistance of edges in $T$. It was also proven by Harvey and Olver in 2014 that the maximum effective resistance of edges in $G$ asymptotically upper bounds $\theta_{G}$. However, finding a mathematical characterization of $\theta_{G}(T)$, even for structured graphs, is still a challenge. In the second part of this thesis, we will give general lower bound and upper bound certificates for $\theta_{G}(T)$ and utilize these certificates for circulant matrices to estimate spectral thinness of graphs such as complete graphs, complete bipartite graphs, and prism graphs.
Nonsmooth Newton Methods for Solving the Best Approximation Problem; with Applications to Linear Programming
(University of Waterloo, 2023-12-19) Weames, Tyler
In this thesis, we study the effects of applying a modified Levenberg-Marquardt regularization to a nonsmooth Newton method. We expand this application to exact and inexact nonsmooth Newton methods and apply it to the best approximation constrained to a polyhedral set problem. We also demonstrate that linear programs can be represented as a best approximation problem, extending the application of nonsmooth Newton methods to linear programming. This application provides us with insight into an external path following algorithm that, like the simplex method, takes a finite number of steps on the boundary of the polyhedral set. However, unlike the simplex method, these steps do not use basic feasible solutions.
Bipartite Quantum Walks and the Hamiltonian
(University of Waterloo, 2023-09-26) Chen, Qiuting
We study a discrete quantum walk model called bipartite walks via a spectral approach. A bipartite walk is determined by a unitary matrix U, i.e., the transition matrix of the walk. For every transition matrix U, there is a Hamiltonian H such that U = exp(iH). If there is a real skew-symmetric matrix S such that H = iS, we say there is a H-digraph associated to the walk and S is the skew-adjacency matrix of the H-digraph. The underlying unweighted non-oriented graph of the H-digraph is H-graph. Let G be a simple bipartite graph with no isolated vertices. The bipartite walk on G is the same as the continuous walk on the H-digraph over integer time. Two questions lie in the centre of this thesis are 1. Is there a connection between the H-(di)graph and the underlying graph G? If there is, what is the connection? 2. Is there a connection between the walk and the underlying graph G? If there is, what is the connection? Given a bipartite walk on G, we show that the underlying bipartite graph G determines the existence of the H-graph. If G is biregular, the spectrum of G determines the spectrum of U. We give complete characterizations of bipartite walks on paths and even cycles. Given a path or an even cycle, the transition matrix of the bipartite walk is a permutation matrix. The H-digraph is an oriented weighted complete graph. Using bipartite walks on even paths, we construct a in nite family of oriented weighted complete graphs such that continuous walks de- ned on them have universal perfect state transfer, which is an interesting but rare phenomenon. Grover's walk is one of the most studied discrete quantum walk model and it can be used to implement the famous Grover's algorithm. We show that Grover's walk is actually a special case of bipartite walks. Moreover, given a bipartite graph G, one step of the bipartite walk on G is the same as two steps of Grover's walk on the same graph. We also study periodic bipartite walks. Using results from algebraic number theory, we give a characterization of periodic walks on a biregular graph with a constraint on its spectrum. This characterization only depends on the spectrum of the underlying graph and the possible spectrum for a periodic walk is determined by the degrees of the underlying graph. We apply this characterization of periodic bipartite walk to Grover's walk to get a characterization of a certain class of periodic Grover's walk. Lastly, we look into bipartite walks on the incidence graphs of incidence structures, t-designs (t 2) and generalized quadrangles in particular. Given a bipartite walk on a t-design, we show that if the underlying design is a partial linear space, the H-graph is the distance-two graph of the line graph of the underlying incidence graph. Given a bipartite walk on the incidence graph of a generalized quadrangle, we show that there is a homogeneous coherent algebra raised from the bipartite walk. This homogeneous coherent algebra is useful in analyzing the behavior of the walk.
Uniform Generation of Graphical Realizations of Joint Degree Matrices
(University of Waterloo, 2023-09-21) Zhou, Qianye
In this thesis, we introduce JDM_GEN, an algorithm designed to uniformly generate graphical realizations of a given joint degree matrix. Amanatidis and Kleer previously employed an MCMC-based method to address this problem. Their method fully resolved the case of two degree classes, and showed that their switch Markov chain is rapidly mixing. While our algorithm imposes certain restrictions on the maximum degrees, it is applicable to any bounded number of degree classes and has a runtime complexity linear in the number of edges.
Rigidity of near-optimal superdense coding protocols
(University of Waterloo, 2023-09-19) Zhou, Xingyu
Rigidity in quantum information theory refers to the stringent constraints underlying optimal or near-optimal performance in certain quantum tasks. This property plays a crucial role in verifying untrusted quantum devices and holds significance for secure quantum protocols. Previous work by Nayak and Yuen demonstrated that all optimal superdense coding protocols are locally equivalent to the canonical Bennett-Wiesner protocol. For higher-dimensional superdense coding protocols, Nayak and Yuen showed they may exist only in a relaxed form, and Farkas, Kaniewski and Nayak showed there are infinitely many dimensions $d\geq 4$ such that the rigidity does not exist even in the relaxed form. Our work is dedicated to establishing the rigidity properties of near-optimal superdense coding protocols. Specifically, we explore scenarios where Alice can employ finite but arbitrary ancilla qubits for encoding, Bob can perform positive operator-valued measure (POVM) for decoding and can answer with error. In such contexts, we prove that any near-optimal superdense coding must be locally equivalent to a superdense coding protocol close to the canonical Bennett-Wiesner protocol. In the search for extending the result to higher dimensional superdense coding protocols, we find a method to orthogonalize any two unitary matrices in the same space. However, the question of whether it is feasible to orthogonalize more than two $d\times d$ unitary matrices when $d>2$ remains an intriguing yet unresolved matter.
Distance-Biregular Graphs and Orthogonal Polynomials
(University of Waterloo, 2023-09-15) Lato, Sabrina
This thesis is about distance-biregular graphs– when they exist, what algebraic and structural properties they have, and how they arise in extremal problems. We develop a set of necessary conditions for a distance-biregular graph to exist. Using these conditions and a computer, we develop tables of possible parameter sets for distancebiregular graphs. We extend results of Fiol, Garriga, and Yebra characterizing distance-regular graphs to characterizations of distance-biregular graphs, and highlight some new results using these characterizations. We also extend the spectral Moore bounds of Cioaba et al. to semiregular bipartite graphs, and show that distance-biregular graphs arise as extremal examples of graphs meeting the spectral Moore bound.
Towards Private Biometric Authentication and Identification
(University of Waterloo, 2023-09-05) Gold, Jonathan
Handwriting and speech are important parts of our everyday lives. Handwriting recognition is the task that allows the recognizing of written text, whether it be letters, words or equations, from given data. When analyzing handwriting, we can analyze static images or the recording of written text through sensors. Handwriting recognition algorithms can be used in many applications, including signature verification, electronic document processing, as well as e-security and e-health related tasks. The OnHW datasets consists of a set of datasets which, through the use of various sensors, captures the writing of characters, words, symbols and equations, recorded in the form of multivariate time series. We begin by developing character recognition models, targeting letters (and later symbols), trained and tested using the OnHW-chars dataset (and later the split OnHW-equations dataset). Our models were able to improve upon the accuracy of the previous best results on both datasets explored. Using our machine learning (ML) models, we provide 11.3%-23.56% improvements over the previous best ML models. Using deep learning (DL), as well as ensemble techniques, we were able to improve on the best previous models by 3.08%-7.01%. In addition to the accuracy improvements, we aim to provide some level of explainability, using a specialized version of LIME for time series data. This explanation helps provide some rationale for why the models make sense for the data, as well as why ensemble methods may be useful to improve accuracy rates for this task. To verify the robustness of our models trained over the OnHW-chars dataset, we trained our DL models using the same model parameters over a more recently published OnHW-equations dataset. Our DL models with ensemble learning provide 0.05%-4.75% improvements over the previous best DL models. While the character recognition task has many applications, when using it to provide a service, it is important to consider user privacy since handwriting is biometric data and contains private information. Next, we design a framework that uses multiparty computation (MPC) to provide users with privacy over their handwritten data, when providing a service for character recognition. We then implement the framework using the models trained on public data to provide private inference on hidden user data. This framework is implemented in the CrypTen MPC framework. We obtain results on the accuracy difference of the models when making inference using MPC, as well as the costs associated with performing this inference. We found a 0.55%-1.42% accuracy difference between plaintext inference and inference with MPC. Next, we pivot to explore writer identification, which involves identifying the writer of some handwritten text. We use the OnHW-equations dataset for our analysis, which at the time of writing has not been used for this task before. We first analyze and reformat the data to fit the writer identification task, as well as remove bias. Using DL models, we obtain accuracy results of up to 91.57% in identifying the writer using their handwriting. As with private inference in the character recognition task, it is important to account for user privacy when training writer identification models and making inference. We design and implement a framework for private training and inference for the writer recognition task, using the CrypTen MPC framework. Since training these models is very costly, we use simpler CNN's for private writer recognition. The chosen CNN trained privately in MPC obtained an accuracy of 77.45%. Next, we analyze the costs associated with privately training the CNN and other CNN's with altered model architectures. Finally, we switch to explore voice as a biometric in the speaker verification task. As with handwriting, a person's voice contains unique characteristics which can be used to determine the speaker. Not only can voice be analyzed similarly with handwriting, in that we can explore the speech recognition and speaker identification tasks, it comes with similar privacy risks for users. We design and implement a unique framework for private speaker verification using the MP-SPDZ MPC framework. We analyze the costs associated with training the model and making inferences, with our main goal being to determine the time it takes to make private inference. We then used these times as part of a survey conducted to determine how much people value the privacy of their biometrics and how long they were willing to wait for the increased privacy. We found that people were willing to tolerate significant time delays in order to privately authenticate themselves, when primed with the benefits of using MPC for privacy.
Enumerating matroid extensions
(University of Waterloo, 2023-09-01) Redlin Hume, Shayla
This thesis investigates the problem of enumerating the extensions of certain matroids. A matroid M is an extension of a matroid N if M delete e is equal to N for some element e of M. Similarly, a matroid M is a coextension of a matroid N if M contract e is equal to N for some element e of M. In this thesis, we consider extensions and coextensions of matroids in the classes of graphic matroids, representable matroids, and frame matroids. We develop a general strategy for counting the extensions of matroids which translates the problem into counting stable sets in an auxiliary graph. We apply this strategy to obtain asymptotic results on the number of extensions and coextensions of certain graphic matroids, projective geometries, and Dowling geometries.
Cryptography and Privacy in Vehicular Communication Networks
(University of Waterloo, 2023-08-28) Sharma, Pravek
Wireless communication technologies can support dynamic networks between vehicles, pedestrians and roadside infrastructure called Vehicular Ad hoc Networks (VANETs). Wireless communication over VANETs allows for several communications scenarios — between vehicles, between vehicles and infrastructure, and between vehicles and pedestrians, among others — collectively known as Vehicle-to-Everything (V2X) communication. Fast wireless communication allows vehicles to communicate over long distances, improving a driver's perception compared to relying on human senses alone. Computerised automated decisions made in response to a wireless message also allow for a lifesaving decision to be much faster than the average human's reaction time can allow. A report by the United Stated Department of Transport shows that applications which use V2X communication, such as Emergency Brake Warning, Left-turn Assist, and Lane-change Assist, can help reduce unimpaired vehicular collisions by as much as 80%. Further, V2X applications like Cooperative Platooning and Emergency Vehicle Path Clearing offer improved fuel efficiency, traffic efficiency, and faster response times for emergency vehicles. For these reasons, V2X communication has garnered significant interest from the automotive industry, the research community and governments in recent years. While V2X communication offers many benefits, unsecured V2X communication can also be exploited by adversaries to increase traffic congestion, track vehicles and people, and even induce vehicular crashes as we show in this thesis. For these reasons, it is necessary to secure VANETs and V2X communication. While security standards for V2X communication exist, their restrictive requirements can make implementing efficient applications difficult. Further, V2X application designers often design applications with little regard to security (incorrectly assuming that the standardised security measures provide adequate security regardless of the underlying application), resulting in applications that violate the security standards imposed restrictions, and leading to applications which are not secure. The Emergency Brake Warning application is one application affected by this disconnect between application designers and V2X security standards. This thesis introduces the uninitiated reader to V2X communication, V2X applications, and V2X security standards while describing the necessary cryptography along the way. Then we discuss the working and limitations of current proposals for the Emergency Brake Warning application before describing EBW-PoF, a novel protocol for the same application, that overcomes these shortcomings. Finally, we discuss EBW-PoF's security, performance, and limitations.
A Linear Algebraic Method on the Chromatic Symmetric Function
(University of Waterloo, 2023-08-28) Haithcock, Evan
The Stanley-Stembridge conjecture is a longstanding conjecture that has evaded proof for nearly 30 years. Concerned with the e-basis expansions of the chromatic symmetric functions of unit-interval graphs, this conjecture has served as a significant motivator of research in algebraic graph theory in recent years. We summarize a great deal of the existing work done in favor of this conjecture, giving an overview of the various techniques that have previously been used in the study of this problem. Moreover, we develop a novel technique using methods from linear algebra and use it to obtain an e-basis expansion of graphs known as single clique-blowups of paths. Using this same method, we use this result to prove the e-positivity of double clique-blowups of paths, a previously unknown result.
Chosen Ciphertext Security from Zero Knowledge Proofs
(University of Waterloo, 2023-08-24) Steckel, Camryn
When designing encryption schemes, there are different levels of security that one can achieve. Of the two main security levels, cryptographers generally strive for the stronger notion of chosen ciphertext attack (CCA) security, which considers attackers who have the ability to obtain decryptions of their choice, over the weaker notion of chosen plaintext attack (CPA) security, which only considers attackers who have encryption abilities. However, it is much easier to find public key encryption schemes (PKEs) that satisfy CPA security. For this reason, a common technique for developing CCA-secure PKEs is to apply a CPA-to-CCA transformation to an existing CPA-secure PKE. The general idea behind such a transform is to somehow ensure that anyone who is capable of producing a valid ciphertext must already know the corresponding plaintext, which renders the additional powers that a CCA adversary has over a CPA adversary entirely useless. All existing transforms achieve this property by performing a re-encryption check in the decryption algorithm. However, this leaves the resulting PKE vulnerable to side-channel attacks, which can be used to carry out chosen ciphertext attacks on the underlying PKE. In this thesis, we present a generic CPA-to-CCA transform that uses a zero-knowledge proof of knowledge in place of a re-encryption check. We prove security of our generic construction in the random oracle model, and we provide an instantiation of it using existing schemes. For the instantiation, we use ElGamal as our underlying PKE, and an application of Fischlin's transfomation to a variant of Schnorr's protocol for our zero-knowledge proof of knowledge, and prove that these protocols satisfy the required security definitions.
Solving Saddle Point Formulations of Linear Programs with Frank-Wolfe
(University of Waterloo, 2023-08-24) Hough, Matthew
The problem of solving a linear program (LP) is ubiquitous in industry, yet in recent years the size of linear programming problems has grown and continues to do so. State-of-the-art LP solvers make use of the Simplex method and primal-dual interior-point methods which are able to provide accurate solutions in a reasonable amount of time for most problems. However, both the Simplex method and interior-point methods require solving a system of linear equations at each iteration, an operation that does not scale well with the size of the problem. In response to the growing size of linear programs and poor scalability of existing algorithms, researchers have started to consider first-order methods for solving large scale linear programs. The best known first-order method for general linear programming problems is PDLP. First-order methods for linear programming are characterized by having a matrix-vector product as their primary computational cost. We present a first-order primal-dual algorithm for solving saddle point formulations of linear programs, named FWLP (Frank-Wolfe Linear Programming). We provide some theoretical results regarding the behavior of our algorithm, however no convergence guarantees are provided. Numerical investigations suggest that our algorithm has error O(1/sqrt(k)) after k iterations, worse than that of PDLP, however we show that our algorithm has advantages for solving very large LPs in practice such as only needing part of the matrix A at each iteration.

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