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

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 ... 
Primal Cutting Plane Methods for the Traveling Salesman Problem
(University of Waterloo, 20170426)Most serious attempts at solving the traveling salesman problem (TSP) are based on the dual fractional cutting plane approach, which moves from one lower bound to the next. This thesis describes methods for implementing ... 
Signature Schemes in the Quantum RandomOracle Model
(University of Waterloo, 20170425)A signature scheme is a fundamental component in modern digital communication. It allows for authenticated messages, without which it would be nearly impossible to ensure security when using most modern technologies. ... 
Digital Signature Schemes Based on Hash Functions
(University of Waterloo, 20170419)Cryptographers and security experts around the world have been awakened to the reality that one day (potentially soon) largescale quantum computers may be available. Most of the publickey cryptosystems employed today on ... 
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 ... 
Efficient Composition of Discrete Time Quantum Walks
(University of Waterloo, 20170120)It is well known that certain search problems are efficiently solved by quantum walk algorithms. Of particular interest are those problems whose efficient solutions involve nesting of search algorithms. The nesting of ... 
On The Density of Binary Matroids Without a Given Minor
(University of Waterloo, 20161221)This thesis is motivated by the following question: how many elements can a simple binary matroid with no $\PG(t,2)$minor have? This is a natural analogue of questions asked about the density of graphs in minorclosed ... 
Structure in Stable Matching Problems
(University of Waterloo, 20161214)In this thesis we provide two contributions to the study of structure in stable matching problems. The first contribution is a short new proof for the integrality of Rothblum’s linear description of the convex hull of ... 
Symmetries
(University of Waterloo, 20161003)Automorphisms of graphs, hypergraphs and disgraphs are investigated. The invariance of the chromatic polynomial in the rotor effect is disproved. New invariance results are obtained. It is shown that given any integer k ... 
On Polynomialtime Pathfollowing Interiorpoint Methods with Local Superlinear Convergence
(University of Waterloo, 20160930)Interiorpoint methods provide one of the most popular ways of solving convex optimization problems. Two advantages of modern interiorpoint methods over other approaches are: (1) robust global convergence, and (2) the ... 
FACES OF MATCHING POLYHEDRA
(University of Waterloo, 20160930)Let G = (V, E, ~) be a finite loopless graph, let b=(bi:ieV) be a vector of positive integers. A feasible matching is a vector X = (x.: j e: E) J of nonnegative integers such that for each node i of G, the sum of ... 
Packing and Covering Odd (u,v)trails in a Graph
(University of Waterloo, 20160927)In this thesis, we investigate the problem of packing and covering odd $(u,v)$trails in a graph. A $(u,v)$trail is a $(u,v)$walk that is allowed to have repeated vertices but no repeated edges. We call a trail \emph{odd} ... 
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 ... 
Planar graphs without 3cycles and with 4cycles far apart are 3choosable
(University of Waterloo, 20160916)A graph G is said to be Lcolourable if for a given list assignment L = {L(v)v ∈ V (G)} there is a proper colouring c of G such that c(v) ∈ L(v) for all v in V (G). If G is Lcolourable for all L with L(v) ≥ k for all ... 
Computing the Residue Class of Partition Numbers
(University of Waterloo, 20160914)In 1919, Ramanujan initiated the study of congruence properties of the integer partition function $p(n)$ by showing that $$p(5n+4) \equiv 0 \mod{5}$$ and $$p(7n+5) \equiv 0 \mod{7}$$ hold for all integers $n$. These results ... 
SUBMODULAR FUNCTIONS, GRAPHS AND INTEGER POLYHEDRA
(University of Waterloo, 20160912)This thesis is a study of the faces of certain combinatorially defined polyhedra. In particular, we examine the vertices and facets of these polyhedra. Chapter 2 contains the essential mathematical background in polyhedral ... 
Covering Graphs and Equiangular Tight Frames
(University of Waterloo, 20160902)Recently, there has been huge attention paid to equiangular tight frames and their constructions, due to the fact that the relationship between these frames and quantum information theory was established. One of the problems ... 
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 ... 
Cyclically 5Connected Graphs
(University of Waterloo, 20160829)Tutte's FourFlow Conjecture states that every bridgeless, Petersenfree graph admits a nowherezero 4flow. This hard conjecture has been open for over half a century with no significant progress in the first forty years. ... 
Unavoidable Minors of Large 5Connected Graphs
(University of Waterloo, 20160824)This thesis shows that, for every positive integer $n \geq 5$, there exists a positive integer $N$ such that every $5$connected graph with at least $N$ vertices has a minor isomorphic to one of thirty explicitly defined ...