Solving Hard Graph Problems with Combinatorial Computing and Optimization

Many problems arising in graph theory are difficult by nature, and finding solutions to large or complex instances of them often require the use of computers. As some such problems are $NP$-hard or lie even higher in the polynomial hierarchy, it is unlikely that efficient, exact algorithms will solve them. Therefore, alternative computational methods are used. Combinatorial computing is a branch of mathematics and computer science concerned with these methods, where algorithms are developed to generate and search through combinatorial structures in order to determine certain properties of them. In this thesis, we explore a number of such techniques, in the hopes of solving specific problem instances of interest. Three separate problems are considered, each of which is attacked with different methods of combinatorial computing and optimization. The first, originally proposed by ErdH{o}s and Hajnal in 1967, asks to find the Folkman number $F_e(3,3;4)$, defined as the smallest order of a $K_4$-free graph that is not the union of two triangle-free graphs. A notoriously difficult problem associated with Ramsey theory, the best known bounds on it prior to this work were $19 leq F_e(3,3;4) leq 941$. We improve the upper bound to $F_e(3,3;4) leq 786$ using a combination of known methods and the Goemans-Williamson semi-definite programming relaxation of MAX-CUT. The second problem of interest is the Ramsey number $R(C_4,K_m)$, which is the smallest $n$ such that any $n$-vertex graph contains a cycle of length four or an independent set of order $m$. With the help of combinatorial algorithms, we determine $R(C_4,K_9)=30$ and $R(C_4,K_{10})=36$ using large-scale computations on the Open Science Grid. Finally, we explore applications of the well-known Lenstra-Lenstra-Lov\u27{a}sz (LLL) algorithm, a polynomial-time algorithm that, when given a basis of a lattice, returns a basis for the same lattice with relatively short vectors. The main result of this work is an application to graph domination, where certain hard instances are solved using this algorithm as a heuristic

Procedures for dealing with certain types of noise and systematic errors common to many Hadamard transform optical systems

Sources of noise and error correcting procedures characteristic of Hadamard transform optical systems were investigated. Reduction of spectral noise due to noise spikes in the data, the effect of random errors, the relative performance of Fourier and Hadamard transform spectrometers operated under identical detector-noise-limited conditions, and systematic means for dealing with mask defects are among the topics discussed. The distortion in Hadamard transform optical instruments caused by moving Masks, incorrect mask alignment, missing measurements, and diffraction is analyzed and techniques for reducing or eliminating this distortion are described

Variational models for color image processing in the RGB space inspired by human vision Mémoire d'Habilitation a Diriger des Recherches dans la spécialité Mathématiques

La recherche que j'ai développée jusqu'à maintenant peut être divisée en quatre catégories principales : les modèles variationnels pourla correction de la couleur basée sur la perception humaine, le transfert d'histogrammes, le traitement d'images à haute gammedynamique et les statistiques d'images naturelles en couleur. Les sujets ci-dessus sont très inter-connectés car la couleur est un sujetfortement inter-disciplinaire

A degree 4 sum-of-squares lower bound for the clique number of the Paley graph

We prove that the degree 4 sum-of-squares (SOS) relaxation of the clique number of the Paley graph on a prime number $p$ of vertices has value at least $\Omega(p^{1/3})$. This is in contrast to the widely believed conjecture that the actual clique number of the Paley graph is $O(\mathrm{polylog}(p))$. Our result may be viewed as a derandomization of that of Deshpande and Montanari (2015), who showed the same lower bound (up to $\mathrm{polylog}(p)$ terms) with high probability for the Erd\H{o}s-R\'{e}nyi random graph on $p$ vertices, whose clique number is with high probability $O(\log(p))$. We also show that our lower bound is optimal for the Feige-Krauthgamer construction of pseudomoments, derandomizing an argument of Kelner. Finally, we present numerical experiments indicating that the value of the degree 4 SOS relaxation of the Paley graph may scale as $O(p^{1/2 - \epsilon})$ for some $\epsilon > 0$, and give a matrix norm calculation indicating that the pseudocalibration proof strategy for SOS lower bounds for random graphs will not immediately transfer to the Paley graph. Taken together, our results suggest that degree 4 SOS may break the "$\sqrt{p}$ barrier" for upper bounds on the clique number of Paley graphs, but prove that it can at best improve the exponent from $1/2$ to $1/3$.Comment: 60 pages, 2 figures, 1 tabl

Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Spectral Properties of Structured Kronecker Products and Their Applications

We study certain spectral properties of some fundamental matrix functions of pairs of symmetric matrices. Our study includes eigenvalue inequalities and various interlacing properties of eigenvalues. We also discuss the role of interlacing in inverse eigenvalue problems for structured matrices. Interlacing is the main ingredient of many fundamental eigenvalue inequalities. This thesis also recounts a historical development of the eigenvalue inequalities relating the sum of two matrices to its summands with some recent findings motivated by problems arising in compressed sensing. One of the fundamental matrix functions on pairs of matrices is the Kronecker product. It arises in many fields such as image processing, signal processing, quantum information theory, differential equations and semidefinite optimization. Kronecker products enjoy useful algebraic properties that have proven to be useful in applications. The less-studied symmetric Kronecker product and skew-symmetric Kronecker product (a contribution of this thesis) arise in semidefinite optimization. This thesis focuses on certain interlacing and eigenvalue inequalities of structured Kronecker products in the context of semidefinite optimization. A popular method used in semidefinite optimization is the primal-dual interior point path following algorithms. In this framework, the Jordan-Kronecker products arise naturally in the computation of Newton search direction. This product also appears in many linear matrix equations, especially in control theory. We study the properties of this product and present some nice algebraic relations. Then, we revisit the symmetric Kronecker product and present its counterpart the skew-symmetric Kronecker product with its basic properties. We settle the conjectures posed by Tunçel and Wolkowicz, in 2003, on interlacing properties of eigenvalues of the Jordan-Kronecker product and inequalities relating the extreme eigenvalues of the Jordan-Kronecker product. We disprove these conjectures in general, but we also identify large classes of matrices for which the interlacing properties hold. Furthermore, we present techniques to generate classes of matrices for which these conjectures fail. In addition, we present a generalization of the Jordan-Kronecker product (by replacing the transpose operator with an arbitrary symmetric involution operator). We study its spectral structure in terms of eigenvalues and eigenvectors and show that the generalization enjoys similar properties of the Jordan-Kronecker product. Lastly, we propose a related structure, namely Lie-Kronecker products and characterize their eigenvectors