Combinatorics and Optimization http://hdl.handle.net/10012/9928 2021-09-21T11:28:44Z Thomassen’s 5-Choosability Theorem Extends to Many Faces http://hdl.handle.net/10012/17374 Thomassen’s 5-Choosability Theorem Extends to Many Faces Nevin, Joshua We prove in this thesis that planar graphs can be L-colored, where L is a list-assignment in which every vertex has a 5-list except for a collection of arbitrarily large faces which have 3-lists, as long as those faces are at least a constant distance apart. Such a result is analogous to Thomassen’s 5-choosability proof where arbitrarily many faces, rather than just one face, are permitted to have 3-lists. This result can also be thought of as a stronger form of a conjecture of Albertson which was solved in 2012 and asked whether a planar graph can be 5-list-colored even if it contains distant precolored vertices. Our result has useful applications in proving that drawings with arbitrarily large pairwise far-apart crossing structures are 5-choosable under certain conditions, and we prove one such result at the end of this thesis. 2021-09-10T00:00:00Z Decomposition-based methods for Connectivity Augmentation Problems http://hdl.handle.net/10012/17338 Decomposition-based methods for Connectivity Augmentation Problems Neogi, Rian In this thesis, we study approximation algorithms for Connectivity Augmentation and related problems. In the Connectivity Augmentation problem, one is given a base graph G=(V,E) that is k-edge-connected, and an additional set of edges $L \subseteq V\times V$ that we refer to as links. The task is to find a minimum cost subset of links $F \subseteq L$ such that adding F to G makes the graph (k+1)-edge-connected. We first study a special case when k=1, which is equivalent to the Tree Augmentation problem. We present a breakthrough result by Adjiashvili that gives an approximation algorithm for Tree Augmentation with approximation guarantee below 2, under the assumption that the cost of every link $\ell \in L$ is bounded by a constant. The algorithm is based on an elegant decomposition based method and uses a novel linear programming relaxation called the $\gamma$-bundle LP. We then present a subsequent result by Fiorini, Gross, Konemann and Sanita who give a $3/2+\epsilon$ approximation algorithm for the same problem. This result uses what are known as Chvatal-Gomory cuts to strengthen the linear programming relaxation used by Adjiashvili, and uses results from the theory of binet matrices to give an improved algorithm that is able to attain a significantly better approximation ratio. Next, we look at the special case when k=2. This case is equivalent to what is known as the Cactus Augmentation problem. A recent result by Cecchetto, Traub and Zenklusen give a 1.393-approximation algorithm for this problem using the same decomposition based algorithmic framework given by Adjiashvili. We present a slightly weaker result that uses the same ideas and obtains a $3/2+\epsilon$ approximation ratio for the Cactus Augmentation problem. Next, we take a look at the integrality ratio of the natural linear programming relaxation for Tree Augmentation, and present a result by Nutov that bounds this integrality gap by 28/15. Finally, we study the related Forest Augmentation problem that is a generalization of Tree Augmentation. There is no approximation algorithm for Forest Augmentation known that obtains an approximation ratio below 2. We show that we can obtain a 29/15-approximation algorithm for Forest Augmentation under the assumption that the LP solution is half-integral via a reduction to Tree Augmentation. We also study the structure of extreme points of the natural linear programming relaxation for Forest Augmentation and prove several properties that these extreme points satisfy. 2021-09-03T00:00:00Z Representations of even-cycle and even-cut matroids http://hdl.handle.net/10012/17297 Representations of even-cycle and even-cut matroids Heo, Cheolwon In this thesis, two classes of binary matroids will be discussed: even-cycle and even-cut matroids, together with problems which are related to their graphical representations. Even-cycle and even-cut matroids can be represented as signed graphs and grafts, respectively. A signed graph is a pair $(G,\Sigma)$ where $G$ is a graph and $\Sigma$ is a subset of edges of $G$. A cycle $C$ of $G$ is a subset of edges of $G$ such that every vertex of the subgraph of $G$ induced by $C$ has an even degree. We say that $C$ is even in $(G,\Sigma)$ if $|C \cap \Sigma|$ is even. A matroid $M$ is an even-cycle matroid if there exists a signed graph $(G,\Sigma)$ such that circuits of $M$ precisely corresponds to inclusion-wise minimal non-empty even cycles of $(G,\Sigma)$. A graft is a pair $(G,T)$ where $G$ is a graph and $T$ is a subset of vertices of $G$ such that each component of $G$ contains an even number of vertices in $T$. Let $U$ be a subset of vertices of $G$ and let $D:= delta_G(U)$ be a cut of $G$. We say that $D$ is even in $(G, T)$ if $|U \cap T|$ is even. A matroid $M$ is an even-cut matroid if there exists a graft $(G,T)$ such that circuits of $M$ corresponds to inclusion-wise minimal non-empty even cuts of $(G,T)$.\\ This thesis is motivated by the following three fundamental problems for even-cycle and even-cut matroids with their graphical representations. (a) Isomorphism problem: what is the relationship between two representations? (b) Bounding the number of representations: how many representations can a matroid have? (c) Recognition problem: how can we efficiently determine if a given matroid is in the class? And how can we find a representation if one exists? These questions for even-cycle and even-cut matroids will be answered in this thesis, respectively. For Problem (a), it will be characterized when two $4$-connected graphs $G_1$ and $G_2$ have a pair of signatures $(\Sigma_1, \Sigma_2)$ such that $(G_1, \Sigma_1)$ and $(G_2, \Sigma_2)$ represent the same even-cycle matroids. This also characterize when $G_1$ and $G_2$ have a pair of terminal sets $(T_1, T_2)$ such that $(G_1,T_1)$ and $(G_2,T_2)$ represent the same even-cut matroid. For Problem (b), we introduce another class of binary matroids, called pinch-graphic matroids, which can generate expo\-nentially many representations even when the matroid is $3$-connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a blocking pair. A blocking pair of a signed graph is a pair of vertices such that every odd cycles intersects with at least one of them. We prove that there exists a constant $c$ such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by $c$. An analogous result for even-cut matroids that are not duals of pinch-graphic matroids will be also proven. As an application, we construct algorithms to solve Problem (c) for even-cycle, even-cut matroids. The input matroids of these algorithms are binary, and they are given by a $(0,1)$-matrix over the finite field $\gf(2)$. The time-complexity of these algorithms is polynomial in the size of the input matrix. 2021-08-27T00:00:00Z Algorithm Substitution Attacks: Detecting ASAs Using State Reset and Making ASAs Asymmetric http://hdl.handle.net/10012/17286 Algorithm Substitution Attacks: Detecting ASAs Using State Reset and Making ASAs Asymmetric Hodges, Philip The field of cryptography has made incredible progress in the last several decades. With the formalization of security goals and the methods of provable security, we have achieved many privacy and integrity guarantees in a great variety of situations. However, all guarantees are limited by their assumptions on the model's adversaries. Edward Snowden's revelations of the participation of the National Security Agency (NSA) in the subversion of standardized cryptography have shown that powerful adversaries will not always act in the way that common cryptographic models assume. As such, it is important to continue to expand the capabilities of the adversaries in our models to match the capabilities and intentions of real world adversaries, and to examine the consequences on the security of our cryptography. In this thesis, we study Algorithm Substitution Attacks (ASAs), which are one way to model this increase in adversary capability. In an ASA, an algorithm in a cryptographic scheme Λ is substituted for a subverted version. The goal of the adversary is to recover a secret that will allow them to compromise the security of Λ, while requiring that the attack is undetectable to the users of the scheme. This model was first formally described by Bellare, Paterson, and Rogaway (Crypto 2014), and allows for the possibility of a wide variety of cryptographic subversion techniques. Since their paper, many successful ASAs on various cryptographic primitives and potential countermeasures have been demonstrated. We will address several shortcomings in the existing literature. First, we formalize and study the use of state resets to detect ASAs. While state resets have been considered as a possible detection method since the first papers on ASAs, future works have only informally reasoned about the effect of state resets on ASAs. We show that many published ASAs that use state are detectable with simple practical methods relying on state resets. Second, we add to the study of asymmetric ASAs, where the ability to recover secrets is restricted to the attacker who implemented the ASA. We describe two asymmetric ASAs on symmetric encryption based on modifications to previous ASAs. We also generalize this result, allowing for any symmetric ASA (on any cryptographic scheme) satisfying certain properties to be transformed into an asymmetric ASA. This work demonstrates the broad application of the techniques first introduced by Bellare, Paterson, and Rogaway (Crypto 2014) and Bellare, Jaeger, and Kane (CCS 2015) and reinforces the need for precise definitions surrounding detectability of stateful ASAs. 2021-08-27T00:00:00Z