Convex Algebraic Geometry Approaches to Graph Coloring and Stable Set Problems

The objective of a combinatorial optimization problem is to find an element that maximizes a given function defined over a large and possibly high-dimensional finite set. It is often the case that the set is so large that solving the problem by inspecting all the elements is intractable. One approach to circumvent this issue is by exploiting the combinatorial structure of the set (and possibly the function) and reformulate the problem into a familiar set-up where known techniques can be used to attack the problem. Some common solution methods for combinatorial optimization problems involve formulations that make use of Systems of Linear Equations, Linear Programs (LPs), Semidefinite Programs (SDPs), and more generally, Conic and Semi-algebraic Programs. Although, generality often implies flexibility and power in the formulations, in practice, an increase in sophistication usually implies a higher running time of the algorithms used to solve the problem. Despite this, for some combinatorial problems, it is hard to rule out the applicability of one formulation over the other. One example of this is the Stable Set Problem. A celebrated result of Lovász's states that it is possible to solve (to arbitrary accuracy) in polynomial time the Stable Set Problem for perfect graphs. This is achieved by showing that the Stable Set Polytope of a perfect graph is the projection of a slice of a Positive Semidefinite Cone of not too large dimension. Thus, the Stable Set Problem can be solved with the use of a reasonably sized SDP. However, it is unknown whether one can solve the same problem using a reasonably sized LP. In fact, even for simple classes of perfect graphs, such as Bipartite Graphs, we do not know the right order of magnitude of the minimum size LP formulation of the problem. Another example is Graph Coloring. In 2008 Jesús De Loera, Jon Lee, Susan Margulies and Peter Malkin proposed a technique to solve several combinatorial problems, including Graph Coloring Problems, using Systems of Linear Equations. These systems are obtained by reformulating the decision version of the combinatorial problem with a system of polynomial equations. By a theorem of Hilbert, known as Hilbert's Nullstellensatz, the infeasibility of this polynomial system can be determined by solving a (usually large) system of linear equations. The size of this system is an exponential function of a parameter $d$ that we call the degree of the Nullstellensatz Certificate. Computational experiments of De Loera et al. showed that the Nullstellensatz method had potential applications for detecting non-$3$-colorability of graphs. Even for known hard instances of graph coloring with up to two thousand vertices and tens of thousands of edges the method was useful. Moreover, all of these graphs had very small Nullstellensatz Certificates. Although, the existence of hard non-$3$-colorable graph examples for the Nullstellensatz approach are known, determining what combinatorial properties makes the Nullstellensatz approach effective (or ineffective) is wide open. The objective of this thesis is to amplify our understanding on the power and limitations of these methods, all of these falling into the umbrella of Convex Algebraic Geometry approaches, for combinatorial problems. We do this by studying the behavior of these approaches for Graph Coloring and Stable Set Problems. First, we study the Nullstellensatz approach for graphs having large girth and chromatic number. We show that that every non-$k$-colorable graph with girth $g$ needs a Nullstellensatz Certificate of degree $\Omega(g)$ to detect its non-$k$-colorability. It is our general belief that the power of the Nullstellensatz method is tied with the interplay between local and global features of the encoding polynomial system. If a graph is locally $k$-colorable, but globally non-$k$-colorable, we suspect that it will be hard for the Nullstellensatz to detect the non-$k$-colorability of the graph. Our results point towards that direction. Finally, we study the Stable Set Problem for $d$-regular Bipartite Graphs having no $C_4$, i.e., having no cycle of length four. In 2017 Manuel Aprile \textit{et al.} showed that the Stable Set Polytope of the incidence graph $G_{d-1}$ of a Finite Projective Plane of order $d-1$ (hence, $d$-regular) does not admit an LP formulation with fewer than $\frac{\ln(d)}{d}|E(G_{d-1})|$ facets. Although, we did not manage to improve this lower bound for general $d$-regular graphs, we show that any $4$-regular bipartite graph $G$ having no $C_4$ does not admit an LP formulation with fewer than $|E(G)|$ facets. In addition, we obtain computational results showing the $|E(G)|$ lower bound also holds for the Finite Projective Plane $G_4$, a $5$-regular graph. It is our belief that Aprile et al. bounds can be improved considerably

Spatial discretizations of generic dynamical systems

How is it possible to read the dynamical properties (ie when the time goes to infinity) of a system on numerical simulations? To try to answer this question, we study in this manuscript a model reflecting what happens when the orbits of a discrete time system $f$ (for example an homeomorphism) are computed numerically . The computer working in finite numerical precision, it will replace $f$ by a spacial discretization of $f$, denoted by $f_N$ (where the order $N$ of discretization stands for the numerical accuracy). In particular, we will be interested in the dynamical behaviour of the finite maps $f_N$ for a generic system $f$ and $N$ going to infinity, where generic will be taken in the sense of Baire (mainly among sets of homeomorphisms or $C^1$-diffeomorphisms). The first part of this manuscript is devoted to the study of the dynamics of the discretizations $f_N$, when $f$ is a generic conservative/dissipative homeomorphism of a compact manifold. We show that it would be mistaken to try to recover the dynamics of $f$ from that of a single discretization $f_N$ : its dynamics strongly depends on the order $N$. To detect some dynamical features of $f$, we have to consider all the discretizations $f_N$ when $N$ goes through $\mathbf N$. The second part deals with the linear case, which plays an important role in the study of $C^1$-generic diffeomorphisms, discussed in the third part of this manuscript. Under these assumptions, we obtain results similar to those established in the first part, though weaker and harder to prove.Comment: 322 pages. This is an improved version of the thesis of the author (among others, the introduction and conclusion have been translated into English). In particular, it contains works already published on arXiv. Comments welcome

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Discrétisations spatiales de systèmes dynamiques génériques

How is it possible to read the dynamical properties (ie when the time goes to infinity) of a system on numerical simulations ? To try to answer this question, we study inthis thesis a model reflecting what happens when the orbits of a discrete time system f (for example an homeomorphism) are computed numerically. The computer working in finite numerical precision, it will replace f by a spacial discretization of f, denotedby f_N (where the order N of discretization stands for the numerical accuracy). In particular, we will be interested in the dynamical behaviour of the finite maps f_N for a generic system f and N going to infinity, where generic will be taken in the sense of Baire (mainly among sets of homeomorphisms or C^1-diffeomorphisms). The first part of this manuscript is devoted to the study of the dynamics of the discretizations f_N, when f is a generic conservative/dissipative homeomorphism of a compact manifold. We show that it would be mistaken to try to recover the dynamics of f from that of a single discretization f_N : its dynamics strongly depends on the order N. To detect some dynamical features of f we have to consider all thediscretizations f_N when N goes through N.The second part deals with the linear case, which plays an important role in the study of C^1-generic diffeomorphisms, discussed in the third part of this manuscript. Under these assumptions, we obtain results similar to those established in the first part,though weaker and harder to prove.Dans quelle mesure peut-on lire les propriétés dynamiques (quand le temps tend vers l’infini) d’un système sur des simulations numériques ? Pour tenter de répondre à cette question, on étudie dans cette thèse un modèle rendant compte de ce qui se passe lorsqu’on calcule numériquement les orbites d’un système à temps discret f (par exemple un homéomorphisme). L’ordinateur travaillant à précision numérique finie, il va remplacer f par une discrétisation spatiale de f, notée f_N (où l’ordre de la discrétisation N rend compte de la précision numérique). On s’intéresse en particulier au comportement dynamique des applications finies f_N pour un système f générique et pour l’ordre N tendant vers l’infini, où générique sera à prendre dans le sens de Baire (principalement parmi des ensembles d’homéomorphismes ou de C^1-difféomorphismes). La première partie de cette thèse est consacrée à l’étude de la dynamique des discrétisations f_N lorsque f est un homéomorphisme conservatif/dissipatif générique d’une variété compacte. L’étude montre qu’il est illusoire de vouloir retrouver la dynamique du système de départ f à partir de celle d’une seule discrétisation f_N : la dynamique de f_N dépend fortement de l’ordre N. Pour détecter certaines dynamiques de f il faut considérer l’ensemble des discrétisations f_N, lorsque N parcourt N.La seconde partie traite du cas linéaire, qui joue un rôle important dans l’étude du cas des C^1-difféomorphismes génériques, abordée dans la troisième partie de cette thèse. Sous ces hypothèses, on obtient des résultats similaires à ceux établis dans la première partie, bien que plus faibles et de preuves plus difficiles

Multispace & Multistructure. Neutrosophic Transdisciplinarity (100 Collected Papers of Sciences), Vol. IV

The fourth volume, in my book series of “Collected Papers”, includes 100 published and unpublished articles, notes, (preliminary) drafts containing just ideas to be further investigated, scientific souvenirs, scientific blogs, project proposals, small experiments, solved and unsolved problems and conjectures, updated or alternative versions of previous papers, short or long humanistic essays, letters to the editors - all collected in the previous three decades (1980-2010) – but most of them are from the last decade (2000-2010), some of them being lost and found, yet others are extended, diversified, improved versions. This is an eclectic tome of 800 pages with papers in various fields of sciences, alphabetically listed, such as: astronomy, biology, calculus, chemistry, computer programming codification, economics and business and politics, education and administration, game theory, geometry, graph theory, information fusion, neutrosophic logic and set, non-Euclidean geometry, number theory, paradoxes, philosophy of science, psychology, quantum physics, scientific research methods, and statistics. It was my preoccupation and collaboration as author, co-author, translator, or cotranslator, and editor with many scientists from around the world for long time. Many topics from this book are incipient and need to be expanded in future explorations