Browsing Combinatorics and Optimization by Title
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2crossing critical graphs with a V8 minor
(University of Waterloo, 20120117)The crossing number of a graph is the minimum number of pairwise crossings of edges among all planar drawings of the graph. A graph G is kcrossing critical if it has crossing number k and any proper subgraph of G has a ... 
5Choosability of Planarplustwoedge Graphs
(University of Waterloo, 20180102)We prove that graphs that can be made planar by deleting two edges are 5choosable. To arrive at this, first we prove an extension of a theorem of Thomassen. Second, we prove an extension of a theorem Postle and Thomas. ... 
Action of degenerate Bethe operators on representations of the symmetric group
(University of Waterloo, 20180524)Degenerate Bethe operators are elements defined by explicit sums in the center of the group algebra of the symmetric group. They are useful on account of their relation to the GelfandZetlin algebra and the YoungJucysMurphy ... 
Acyclic Colouring of Graphs on Surfaces
(University of Waterloo, 20180904)An acyclic kcolouring of a graph G is a proper kcolouring of G with no bichromatic cycles. In 1979, Borodin proved that planar graphs are acyclically 5colourable, an analog of the Four Colour Theorem. Kawarabayashi and ... 
ADMM for SDP Relaxation of GP
(University of Waterloo, 20160830)We consider the problem of partitioning the set of nodes of a graph G into k sets of given sizes in order to minimize the cut obtained after removing the kth set. This is a variant of the wellknown vertex separator ... 
Algebraic Analysis of VertexDistinguishing EdgeColorings
(University of Waterloo, 2006)Vertexdistinguishing edgecolorings (vdec colorings) are a restriction of proper edgecolorings. These special colorings require that the sets of edge colors incident to every vertex be distinct. This is a relatively ... 
Algebraic Aspects of MultiParticle Quantum Walks
(University of Waterloo, 20121204)A continuous time quantum walk consists of a particle moving among the vertices of a graph G. Its movement is governed by the structure of the graph. More formally, the adjacency matrix A is the Hamiltonian that determines ... 
Algebraic Methods and Monotone Hurwitz Numbers
(University of Waterloo, 20120921)We develop algebraic methods to solve joincut equations, which are partial differential equations that arise in the study of permutation factorizations. Using these techniques, we give a detailed study of the recently ... 
Algebraic Methods for Reducibility in NowhereZero Flows
(University of Waterloo, 20070925)We study reducibility for nowherezero flows. A reducibility proof typically consists of showing that some induced subgraphs cannot appear in a minimum counterexample to some conjecture. We derive algebraic proofs of ... 
Algebraic Tori in Cryptography
(University of Waterloo, 2005)Communicating bits over a network is expensive. Therefore, cryptosystems that transmit as little data as possible are valuable. This thesis studies several cryptosystems that require significantly less bandwidth than ... 
Analyzing Quantum Cryptographic Protocols Using Optimization Techniques
(University of Waterloo, 20120522)This thesis concerns the analysis of the unconditional security of quantum cryptographic protocols using convex optimization techniques. It is divided into the study of coinflipping and oblivious transfer. We first examine ... 
Applications of Bilinear Maps in Cryptography
(University of Waterloo, 2002)It was recently discovered by Joux [30] and Sakai, Ohgishi and Kasahara [47] that bilinear maps could be used to construct cryptographic schemes. Since then, bilinear maps have been used in applications as varied as ... 
Applications of Semidefinite Programming in Quantum Cryptography
(University of Waterloo, 20070518)Coinflipping is the cryptographic task of generating a random coinflip between two mistrustful parties. Kitaev discovered that the security of quantum coinflipping protocols can be analyzed using semidefinite programming. ... 
Applied Hilbert's Nullstellensatz for Combinatorial Problems
(University of Waterloo, 20160923)Various feasibility problems in Combinatorial Optimization can be stated using systems of polynomial equations. Determining the existence of a \textit{stable set} of a given size, finding the \textit{chromatic number} of ... 
Approximate Private Quantum Channels
(University of Waterloo, 2006)This thesis includes a survey of the results known for private and approximate private quantum channels. We develop the best known upper bound for εrandomizing maps, <em>n</em> + 2log(1/ε) + <em>c</em> ... 
Approximating MinimumSize 2EdgeConnected and 2VertexConnected Spanning Subgraphs
(University of Waterloo, 20170427)We study the unweighted 2edgeconnected and 2vertexconnected spanning subgraph problems. A graph is 2edgeconnected if it is connected on removal of an edge, and it is 2vertexconnected if it is connected on removal ... 
Approximation Algorithms for (S,T)Connectivity Problems
(University of Waterloo, 20100803)We study a directed network design problem called the $k$$(S,T)$connectivity problem; we design and analyze approximation algorithms and give hardness results. For each positive integer $k$, the minimum cost $k$vertex ... 
Approximation Algorithms for Clustering and Facility Location Problems
(University of Waterloo, 20170406)Facility location problems arise in a wide range of applications such as plant or warehouse location problems, cache placement problems, and network design problems, and have been widely studied in Computer Science and ... 
Approximation Algorithms for Graph Protection Problems
(University of Waterloo, 20150522)We study a budgeted cut problem known as Graph Protection, where the goal is to remove edges of a given graph in order to protect valuable nodes from stochastic, infectious threats. This problem was recently proposed ... 
Approximation Algorithms for Path TSP, ATSP, and TAP via Relaxations
(University of Waterloo, 20150729)Linear programming (LP) relaxations provide a powerful technique to design approximation algorithms for combinatorial optimization problems. In the first part of the thesis, we study the metric st path Traveling Salesman ...