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Welcome to the Department of Combinatorics and Optimization sub-community.

Research is organized into these collections:

• Combinatorics and Optimization Department: Faculty, student, and staff research arranged by type. This collection does not include graduate Theses & Dissertations.
• Combinatorics and Optimization Theses & Dissertations: Graduate student research required for degree completion.

Recent Submissions

• Efficient Composition of Discrete Time Quantum Walks ﻿

(University of Waterloo, 2017-01-20)
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, 2016-12-21)
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 minor-closed ...
• Structure in Stable Matching Problems ﻿

(University of Waterloo, 2016-12-14)
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, 2016-10-03)
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 Polynomial-time Path-following Interior-point Methods with Local Superlinear Convergence ﻿

(University of Waterloo, 2016-09-30)
Interior-point methods provide one of the most popular ways of solving convex optimization problems. Two advantages of modern interior-point methods over other approaches are: (1) robust global convergence, and (2) the ...
• FACES OF MATCHING POLYHEDRA ﻿

(University of Waterloo, 2016-09-30)
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, 2016-09-27)
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, 2016-09-23)
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 3-cycles and with 4-cycles far apart are 3-choosable ﻿

(University of Waterloo, 2016-09-16)
A graph G is said to be L-colourable 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 L-colourable for all L with |L(v)| ≥ k for all ...
• Computing the Residue Class of Partition Numbers ﻿

(University of Waterloo, 2016-09-14)
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, 2016-09-12)
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, 2016-09-02)
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, 2016-08-30)
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 k-th set. This is a variant of the well-known vertex separator ...
• Cyclically 5-Connected Graphs ﻿

(University of Waterloo, 2016-08-29)
Tutte's Four-Flow Conjecture states that every bridgeless, Petersen-free graph admits a nowhere-zero 4-flow. This hard conjecture has been open for over half a century with no significant progress in the first forty years. ...
• Unavoidable Minors of Large 5-Connected Graphs ﻿

(University of Waterloo, 2016-08-24)
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 ...
• On the Strongly Connected Components of Random Directed Graphs with Given Degree Sequences ﻿

(University of Waterloo, 2016-08-24)
A strongly connected component of a directed graph G is a maximal subgraph H of G such that for each pair of vertices u and v in H, there is a directed path from u to v and a directed path from v to u in H. A strongly ...
• On the effectiveness of isogeny walks for extending cover attacks on elliptic curves ﻿

(University of Waterloo, 2016-08-23)
Cryptographic systems based on the elliptic curve discrete logarithm problem (ECDLP) are widely deployed in the world today. In order for such a system to guarantee a particular security level, the elliptic curve selected ...
• A Study of Time Representation in a Class of Short Term Scheduling Problems ﻿

(University of Waterloo, 2016-08-17)
The problem of scheduling operations has received significant attention from academia and industrial practitioners in the past few decades. A key decision in various scheduling operations problems is when to perform an ...
• Combinatorial Methods for Enumerating Maps in Surfaces of Arbitrary Genus ﻿

(University of Waterloo, 2016-06-10)
The problem of map enumeration is one that has been studied intensely for the past half century. Early work on this subject included the works of Tutte for various types of rooted planar maps and the works of Brown for ...
• Matchings and Covers in Hypergraphs ﻿

(University of Waterloo, 2016-05-17)
In this thesis, we study three variations of matching and covering problems in hypergraphs. The first is motivated by an old conjecture of Ryser which says that if $\mcH$ is an $r$-uniform, $r$-partite hypergraph which ...

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