Combinatorics and Optimization

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

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Differential Equations and Depth First Search for Enumeration of Maps in Surfaces
(University of Waterloo, 1999) Brown, Daniel
A map is an embedding of the vertices and edges of a graph into a compact 2-manifold such that the remainder of the surface has components homeomorphic to open disks. With the goal of proving the Four Colour Theorem, Tutte began the field of map enumeration in the 1960's. His methods included developing the edge deletion decomposition, developing and solving a recurrence and functional equation based on this decomposition, and developing the medial bijection between two equinumerous infinite families of maps. Beginning in the 1980's Jackson, Goulden and Visentin applied algebraic methods in enumeration of non-planar and non-orientable maps, to obtain results of interest for mathematical physics and algebraic geometry, and the Quadrangulation Conjecture and the Map-Jack Conjecture. A special case of the former is solved by Tutte's medial bijection. The latter uses Jack symmetric functions which are a topic of active research. In the 1960's Walsh and Lehman introduced a method of encoding orientable maps. We develop a similar method, based on depth first search and extended to non-orientable maps. With this, we develop a bijection that extends Tutte's medial bijection and partially solves the Quadrangulation Conjecture. Walsh extended Tutte's recurrence for planar maps to a recurrence for all orientable maps. We further extend the recurrence to include non-orientable maps, and express it as a partial differential equation satisfied by the generating series. By appropriately interpolating the differential equation and applying the depth first search method, we construct a parameter that empirically fulfils the conditions of the Map-Jack Conjecture, and we prove some of its predicted properties. Arques and Beraud recently obtained a continued fraction form of a specialisation of the generating series for maps. We apply the depth search method with an ordinary differential equation, to construct a bijection whose existence is implied by the continued fraction.
A survey on Traitor Tracing Schemes
(University of Waterloo, 2000) Chen, Jason
When intellectual properties are distributed over a broadcast network, the content is usually encrypted in a way such that only authorized users who have a certain set of keys, can decrypt the content. Some authorized users may be willing to disclose their keys in constructing a pirate decoder which allows illegitimate users to access the content. It is desirable to determine the source of the keys in a pirate decoder, once one is captured. Traitor tracing schemes were introduced to help solve this problem. A traitor tracing scheme usually consists of: a scheme to generate and distribute each user's personal key, a cryptosystem used to protect session keys that are used to encrypt/decrypt the actual content, and a tracing algorithm to determine one source of the keys in a pirate decoder. In this thesis, we survey the traitor tracing schemes that have been suggested. We group the schemes into two groups: symmetric in which the session key is encrypted and decrypted using the same key and asymmetric schemes in which the session key is encrypted and decrypted using different keys. We also explore the possibility of a truly public scheme in which the data supplier knows the encryption keys only. A uniform analysisis presented on the efficiency of these schemes using a set of performance parameters.
Matrix Formulations of Matching Problems
(University of Waterloo, 2000) Webb, Kerri
Finding the maximum size of a matching in an undirected graph and finding the maximum size of branching in a directed graph can be formulated as matrix rank problems. The Tutte matrix, introduced by Tutte as a representation of an undirected graph, has rank equal to the maximum number of vertices covered by a matching in the associated graph. The branching matrix, a representation of a directed graph, has rank equal to the maximum number of vertices covered by a branching in the associated graph. A mixed graph has both undirected and directed edges, and the matching forest problem for mixed graphs, introduced by Giles, is a generalization of the matching problem and the branching problem. A mixed graph can be represented by the matching forest matrix, and the rank of the matching forest matrix is related to the size of a matching forest in the associated mixed graph. The Tutte matrix and the branching matrix have indeterminate entries, and we describe algorithms that evaluate the indeterminates as rationals in such a way that the rank of the evaluated matrix is equal to the rank of the indeterminate matrix. Matroids in the context of graphs are discussed, and matroid formulations for the matching, branching, and matching forest problems are given.
A survey of the trust region subproblem within a semidefinite framework
(University of Waterloo, 2000) Fortin, Charles
Trust region subproblems arise within a class of unconstrained methods called trust region methods. The subproblems consist of minimizing a quadratic function subject to a norm constraint. This thesis is a survey of different methods developed to find an approximate solution to the subproblem. We study the well-known method of More and Sorensen and two recent methods for large sparse subproblems: the so-called Lanczos method of Gould et al. and the Rendland Wolkowicz algorithm. The common ground to explore these methods will be semidefinite programming. This approach has been used by Rendl and Wolkowicz to explain their method and the More and Sorensen algorithm; we extend this work to the Lanczos method. The last chapter of this thesis is dedicated to some improvements done to the Rendl and Wolkowicz algorithm and to comparisons between the Lanczos method and the Rendl and Wolkowicz algorithm. In particular, we show some weakness of the Lanczos method and show that the Rendl and Wolkowicz algorithm is more robust.
Variational Spectral Analysis
(University of Waterloo, 2000) Sendov, Hristo
We present results on smooth and nonsmooth variational properties of {it symmetric} functions of the eigenvalues of a real symmetric matrix argument, as well as {it absolutely symmetric} functions of the singular values of a real rectangular matrix. Such results underpin the theory of optimization problems involving such functions. We answer the question of when a symmetric function of the eigenvalues allows a quadratic expansion around a matrix, and then the stronger question of when it is twice differentiable. We develop simple formulae for the most important nonsmooth subdifferentials of functions depending on the singular values of a real rectangular matrix argument and give several examples. The analysis of the above two classes of functions may be generalized in various larger abstract frameworks. In particular, we investigate how functions depending on the eigenvalues or the singular values of a matrix argument may be viewed as the composition of symmetric functions with the roots of {it hyperbolic polynomials}. We extend the relationship between hyperbolic polynomials and {it self-concordant barriers} (an extremely important class of functions in contemporary interior point methods for convex optimization) by exhibiting a new class of self-concordant barriers obtainable from hyperbolic polynomials.
New Convex Relaxations for the Maximum Cut and VLSI Layout Problems
(University of Waterloo, 2001) Ferreira Fialho dos Anjos, Miguel Nuno
It is well known that many of the optimization problems which arise in applications are hard, which usually means that they are NP-hard. Hence much research has been devoted to finding good relaxations for these hard problems. Usually a good relaxation is one which can be solved (either exactly or within a prescribed numerical tolerance) in polynomial-time. Nesterov and Nemirovskii showed that by this criterion, many convex optimization problems are good relaxations. This thesis presents new convex relaxations for two such hard problems, namely the Maximum-Cut (Max-Cut) problem and the VLSI (Very Large Scale Integration of electronic circuits) layout problem. We derive and study the properties of two new strengthened semidefinite programming relaxations for the Max-Cut problem. Our theoretical results hold for every instance of Max-Cut; in particular, we make no assumptions about the edge weights. The first relaxation provides a strengthening of the well-known Goemans-Williamson relaxation, and the second relaxation is a further tightening of the first. We prove that the tighter relaxation automatically enforces the well-known triangle inequalities, and in fact is stronger than the simple addition of these inequalities to the Goemans-Williamson relaxation. We further prove that the tighter relaxation fully characterizes some low dimensional faces of the cut polytope via the rank of its feasible matrices. We also address some practical issues arising in the solution of these relaxations and present numerical results showing the remarkably good bounds computed by the tighter relaxation. For the VLSI layout problem, we derive a new relaxation by extending the target distance concept recently introduced by Etawil and Vannelli. The resulting AR (Attractor-Repeller) model improves on the NLT (Non-linear optimization Layout Technique) method of van Camp et al. by efficiently finding a good initial point for the NLT solver. Because the AR model is not a convex optimization problem, we isolate the source of the non-convexity and thereby derive a convex version of the model. This derivation leads to the definition of certain generalized target distances with an interesting practical interpretation. Finally, we present numerical results that demonstrate the potential of the AR model.
Techniques of Side Channel Cryptanalysis
(University of Waterloo, 2001) Muir, James
The traditional model of cryptography examines the security of cryptographic primitives as mathematical functions. This approach does not account for the physical side effects of using these primitives in the real world. A more realistic model employs the concept of a side channel. A side channel is a source of information that is inherent to a physical implementation of a primitive. Research done in the last half of the 1990s has shown that the information transmitted by side channels, such as execution time, computational faults and power consumption, can be detrimental to the security of ciphers like DES and RSA. This thesis surveys the techniques of side channel cryptanalysis presented in [Kocher1996], [Boneh1997], and [Kocher1998] and shows how side channel information can be used to break implementations of DES and RSA. Some specific techniques covered include the timing attack, differential fault analysis, simple power analysis and differential power analysis. Possible defenses against each of these side channel attacks are also discussed.
Parking Functions and Related Combinatorial Structures.
(University of Waterloo, 2001) Rattan, Amarpreet
The central topic of this thesis is parking functions. We give a survey of some of the current literature concerning parking functions and focus on their interaction with other combinatorial objects; namely noncrossing partitions, hyperplane arrangements and tree inversions. In the final chapter, we discuss generalizations of both parking functions and the above structures.
The Existence of Balanced Tournament Designs and Partitioned Balanced Tournament Designs
(University of Waterloo, 2001) Bauman, Shane
A balanced tournament design of order n, BTD(n), defined on a 2n-set V, is an arrangement of the all of the (2n2) distinct unordered pairs of elements of V into an n X (2n - 1) array such that (1) every element of V occurs exactly once in each column and (2) every element of V occurs at most twice in each row. We will show that there exists a BTD(n) for n a positive integer, n not equal to 2. For n = 2, a BTD (n) does not exist. If the BTD(n) has the additional property that it is possible to permute the columns of the array such that for every row, all the elements of V appear exactly once in the first n pairs of that row and exactly once in the last n pairs of that row then we call the design a partitioned balanced tournament design, PBTD(n). We will show that there exists a PBTD (n) for n a positive integer, n is greater than and equal to 5, except possibly for n an element of the set {9,11,15}. For n less than and equal to 4 a PBTD(n) does not exist.
Hamilton Paths in Generalized Petersen Graphs
(University of Waterloo, 2002) Pensaert, William
This thesis puts forward the conjecture that for n > 3k with k > 2, the generalized Petersen graph, GP(n,k) is Hamilton-laceable if n is even and k is odd, and it is Hamilton-connected otherwise. We take the first step in the proof of this conjecture by proving the case n = 3k + 1 and k greater than or equal to 1. We do this mainly by means of an induction which takes us from GP(3k + 1, k) to GP(3(k + 2) + 1, k + 2). The induction takes the form of mapping a Hamilton path in the smaller graph piecewise to the larger graph an inserting subpaths we call rotors to obtain a Hamilton path in the larger graph.
Convergence Analysis of Generalized Primal-Dual Interior-Point Algorithms for Linear Optimization
(University of Waterloo, 2002) Wei, Hua
We study the zeroth-, first-, and second-order algorithms proposed by Tuncel. The zeroth-order algorithms are the generalization of the classic primal-dual affine-scaling methods, and have a strong connection with the quasi-Newton method. Although the zeroth-order algorithms have the property of strict monotone decrease in both primal and dual objective values, they may not converge. We give an illustrative example as well as an algebraic proof to show that the zeroth-order algorithms do not converge to an optimal solution in some cases. The second-order algorithms use the gradients and Hessians of the barrier functions. Tuncel has shown that all second-order algorithms have a polynomial iteration bound. The second-order algorithms have a range of primal-dual scaling matrices to be chosen. We give a method to construct such a primal-dual scaling matrix. We then analyze a new centrality measure. This centrality measure appeared in both first- and second-order algorithms. We compare the neighbourhood defined by this centrality measure with other known neighbourhoods. We then analyze how this centrality measure changes in the next iteration in terms of the step length and some other information of the current iteration.
Applications of Bilinear Maps in Cryptography
(University of Waterloo, 2002) Gagne, Martin
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 identity-based encryption, short signatures and one-round tripartite key agreement. This thesis explains the notion of bilinear maps and surveys the applications of bilinear maps in the three main fields of cryptography: encryption, signature and key agreement. We also show how these maps can be constructed using the Weil and Tate pairings in elliptic curves.
Digital Signature Scheme Variations
(University of Waterloo, 2002) Dunbar, Fiona
A digital signature scheme is the process of signing an electronic message that can be transmitted over a computer network. Digital signatures provide message authentication that can be proved to a third party. With the rise of electronic communications over the Internet, digital signatures are becoming increasingly important, especially for the exchange of messages of legal significance. In 1988, Goldwasser, Micali and Rivest (GMR) [31] defined a signature scheme as a collection of algorithms: key generation, signature generation and signature verification. They defined a signature scheme as secure if it was existentially unforgeable against a chosen-message attack. These general definitions suited most signatures at the time, however, over the last decade digital signatures have emerged for which the GMR definitions are unsuitable. These signature schemes, together with their applications and security and efficiency considerations, will be explored in this thesis. These signature scheme variations have been classified by the additional services they provide to ordinary signature schemes, namely increased efficiency, increased security, anonymity, and enhanced signing and verifying capabilities.
Self-Dual Graphs
(University of Waterloo, 2002) Hill, Alan
The study of self-duality has attracted some attention over the past decade. A good deal of research in that time has been done on constructing and classifying all self-dual graphs and in particular polyhedra. We will give an overview of the recent research in the first two chapters. In the third chapter, we will show the necessary condition that a self-complementary self-dual graph have n ≡ 0, 1 (mod 8) vertices and we will review White's infinite class (the Paley graphs, for which n ≡ 1 (mod 8)). Finally, we will construct a new infinite class of self-complementary self-dual graphs for which n ≡ 0 (mod 8).
Continuous-time Quantum Algorithms: Searching and Adiabatic Computation
(University of Waterloo, 2002) Ioannou, Lawrence
One of the most important quantum algorithms is Grover's search algorithm [G96]. Quantum searching can be used to speed up the search for solutions to NP-complete problems e. g. 3SAT. Even so, the best known quantum algorithms for 3SAT are considered inefficient. Soon after Grover's discovery, Farhi and Gutmann [FG96] devised a "continuous-time analogue" of quantum searching. More recently Farhi et. al. [FGGS00] proposed a continuous-time 3SAT algorithm which invokes the adiabatic approximation [M76]. Their algorithm is difficult to analyze, hence we do not know whether it can solve typical 3SAT instances faster than Grover's search algorithm can. I begin with a review of the discrete- and continuous-time models of quantum computation. I then make precise the notion of "efficient quantum algorithms", motivating sufficient conditions for discrete- and continuous-time algorithms to be considered efficient via discussion of standard techniques for discrete-time simulation of continuous-time algorithms. After reviewing three quantum search algorithms [F00,FG96,G96], I develop the adiabatic 3SAT algorithm as a natural extension of Farhi and Gutmann's search algorithm. Along the way, I present the adiabatic search algorithm [vDMV01] and remark on its discrete-time simulation. Finally I devise a generalization of the adiabatic algorithm and prove some lower bounds for various cases of this general framework. UPDATE (February 2003): Please see article http://arxiv. org/abs/quant-ph/0302138 for a resolution to the problem of simulating the continuous-time adiabatic search algorithm with a quantum circuit using only O(sqrt(N)) resources.
Uniqueness and Complexity in Generalised Colouring
(University of Waterloo, 2003) Farrugia, Alastair
The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition of a family of graphs that are colourable according to certain rules. In this thesis, we study additive induced-hereditary families, and some generalisations, from a colouring perspective. Our main results are: · Additive induced-hereditary families are uniquely factorisable into irreducible families. · If P and Q are additive induced-hereditary graph families, then (P,Q)-COLOURING is NP-hard, with the exception of GRAPH 2-COLOURING. Moreover, with the same exception, (P,Q)-COLOURING is NP-complete iff P- and Q-RECOGNITION are both in NP. This proves a 1997 conjecture of Kratochvíl and Schiermeyer. We also provide generalisations to somewhat larger families. Other results that we prove include: · a characterisation of the minimal forbidden subgraphs of a hereditary property in terms of its minimal forbidden induced-subgraphs, and vice versa; · extensions of Mihók's construction of uniquely colourable graphs, and Scheinerman's characterisations of compositivity, to disjoint compositive properties; · an induced-hereditary property has at least two factorisations into arbitrary irreducible properties, with an explicitly described set of exceptions; · if G is a generating set for A ο B, where A and B are indiscompositive, then we can extract generating sets for A and B using a greedy algorithm.
An Optimizing Pulse Sequence Compiler for NMR QIP
(University of Waterloo, 2003) Perez Delgado, Carlos Antonio
Quantum information processing is a multi-disciplinary science involving physics, mathematics, computer science, and even quantum chemistry. It is centred around the idea of manipulating physical systems at the quantum level, either for simulation of physical systems, or numerical computation. Although it has been known for almost a decade that a quantum computer would enable the solution of problems deemed infeasible classically, constructing one has been beyond today's capabilities. In this work we explore one proposed implementation of a quantum computer: Nuclear Magnetic Resonance (NMR) spectroscopy. We also develop a numerical software tool, a pulse sequence compiler, for use in the implementation of quantum computer programs on an NMR quantum computer. Our pulse sequence compiler takes as input the specifications of the molecule used as a quantum register, the desired quantum gate, and experimental data on the actual effects of RF pulses on a sample of the molecule, and outputs an optimum set of pre and post 'virtual' gates that minimize the error induced.
Perfect Hash Families: Constructions and Applications
(University of Waterloo, 2003) Kim, Kyung-Mi
Let A and B be finite sets with |A|=n and |B|=m. An (n,m,w)-perfect hash family is a collection F of functions from A to B such that for any X ⊆ A with |X|=w, there exists at least one ? ∈ F such that ? is one-to-one when restricted to X. Perfect hash families are basic combinatorial structures and they have played important roles in Computer Science in areas such as database management, operating systems, and compiler constructions. Such hash families are used for memory efficient storage and fast retrieval of items such as reserved words in programming languages, command names in interactive systems, or commonly used words in natural languages. More recently, perfect hash families have found numerous applications to cryptography, for example, to broadcast encryption schemes, secret sharing, key distribution patterns, visual cryptography, cover-free families and secure frameproof codes. In this thesis, we survey constructions and applications of perfect hash families. For constructions, we divided the results into three parts, depending on underlying structure and properties of the constructions: combinatorial structures, linear functionals, and algebraic structures. For applications, we focus on those related to cryptography.
Postman Problems on Mixed Graphs
(University of Waterloo, 2003) Zaragoza Martinez, Francisco Javier
The mixed postman problem consists of finding a minimum cost tour of a mixed graph M = (V,E,A) traversing all its edges and arcs at least once. We prove that two well-known linear programming relaxations of this problem are equivalent. The extra cost of a mixed postman tour T is the cost of T minus the cost of the edges and arcs of M. We prove that it is NP-hard to approximate the minimum extra cost of a mixed postman tour. A related problem, known as the windy postman problem, consists of finding a minimum cost tour of an undirected graph G=(V,E) traversing all its edges at least once, where the cost of an edge depends on the direction of traversal. We say that G is windy postman perfect if a certain windy postman polyhedron O (G) is integral. We prove that series-parallel undirected graphs are windy postman perfect, therefore solving a conjecture of Win. Given a mixed graph M = (V,E,A) and a subset R ⊆ E ∪ A, we say that a mixed postman tour of M is restricted if it traverses the elements of R exactly once. The restricted mixed postman problem consists of finding a minimum cost restricted tour. We prove that this problem is NP-hard even if R=A and we restrict M to be planar, hence solving a conjecture of Veerasamy. We also prove that it is NP-complete to decide whether there exists a restricted tour even if R=E and we restrict M to be planar. The edges postman problem is the special case of the restricted mixed postman problem when R=A. We give a new class of valid inequalities for this problem. We introduce a relaxation of this problem, called the b-join problem, that can be solved in polynomial time. We give an algorithm which is simultaneously a 4/3-approximation algorithm for the edges postman problem, and a 2-approximation algorithm for the extra cost of a tour. The arcs postman problem is the special case of the restricted mixed postman problem when R=E. We introduce a class of necessary conditions for M to have an arcs postman tour, and we give a polynomial-time algorithm to decide whether one of these conditions holds. We give linear programming formulations of this problem for mixed graphs arising from windy postman perfect graphs, and mixed graphs whose arcs form a forest.
Regularization Using a Parameterized Trust Region Subproblem
(University of Waterloo, 2004) Grodzevich, Oleg
We present a new method for regularization of ill-conditioned problems that extends the traditional trust-region approach. Ill-conditioned problems arise, for example, in image restoration or mathematical processing of medical data, and involve matrices that are very ill-conditioned. The method makes use of the L-curve and L-curve maximum curvature criterion as a strategy recently proposed to find a good regularization parameter. We describe the method and show its application to an image restoration problem. We also provide a MATLAB code for the algorithm. Finally, a comparison to the CGLS approach is given and analyzed, and future research directions are proposed.

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