Extension complexity of stable set polytopes of bipartite graphs

The extension complexity $\mathsf{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that affinely projects to $P$. Let $G$ be a bipartite graph with $n$ vertices, $m$ edges, and no isolated vertices. Let $\mathsf{STAB}(G)$ be the convex hull of the stable sets of $G$. It is easy to see that $n \leqslant \mathsf{xc} (\mathsf{STAB}(G)) \leqslant n+m$. We improve both of these bounds. For the upper bound, we show that $\mathsf{xc} (\mathsf{STAB}(G))$ is $O(\frac{n^2}{\log n})$, which is an improvement when $G$ has quadratically many edges. For the lower bound, we prove that $\mathsf{xc} (\mathsf{STAB}(G))$ is $\Omega(n \log n)$ when $G$ is the incidence graph of a finite projective plane. We also provide examples of $3$-regular bipartite graphs $G$ such that the edge vs stable set matrix of $G$ has a fooling set of size $|E(G)|$.Comment: 13 pages, 2 figure

Polütoopide laienditega seotud ülesanded

Väitekirja elektrooniline versioon ei sisalda publikatsiooneLineaarplaneerimine on optimeerimine matemaatilise mudeliga, mille sihi¬funktsioon ja kitsendused on esitatud lineaarsete seostega. Paljusid igapäeva elu väljakutseid võime vaadelda lineaarplaneerimise vormis, näiteks miinimumhinna või maksimaalse tulu leidmist. Sisepunkti meetod saavutab häid tulemusi nii teoorias kui ka praktikas ning lahendite leidmise tööaeg ja lineaarsete seoste arv on polünomiaalses seoses. Sellest tulenevalt eksponentsiaalne arv lineaarseid seoseid väljendub ka ekponentsiaalses tööajas. Iga vajalik lineaarne seos vastab ühele polütoobi P tahule, mis omakorda tähistab lahendite hulka. Üks võimalus tööaja vähendamiseks on suurendada dimensiooni, mille tulemusel väheneks ka polütoobi tahkude arv. Saadud polütoopi Q nimeta¬takse polütoobi P laiendiks kõrgemas dimensioonis ning polütoobi Q minimaalset tahkude arvu nimetakakse polütoobi P laiendi keerukuseks, sellisel juhul optimaalsete lahendite hulk ei muutu. Tekib küsimus, millisel juhul on võimalik leida laiend Q, mille korral tahkude arv on polünomiaalne. Mittedeterministlik suhtluskeerukus mängib olulist rolli tõestamaks polütoopide laiendite keerukuse alampiiri. Polütoobile P vastava suhtluskeerukuse leidmine ning alamtõkke tõestamine väistavad võimalused leida laiend Q, mis ei oleks eksponentsiaalne. Käesolevas töös keskendume me juhuslikele Boole'i funktsioonidele f, mille tihedusfunktsioon on p = p(n). Me pakume välja vähima ülemtõkke ning suurima alamtõkke mittedeterministliku suhtluskeerukuse jaoks. Lisaks uurime me ka pedigree polütoobi graafi. Pedigree polütoop on rändkaupmehe ülesande polütoobi laiend, millel on kombinatoorne struktuur. Polütoobi graafi võib vaadelda kui abstraktset graafi ning see annab informatsiooni polütoobi omaduste kohta.The linear programming (LP for short) is a method for finding an optimal solution, such as minimum cost or maximum profit for a linear function subject to linear constraints. But having an exponential number of inequalities gives the exponential running time in solving linear program. A polytope, let's say P, represents the space of the feasible solution. One idea for decreasing the running time of the problem, is lifting the polytope P tho the higher dimensions with the goal of decresing the number of inequalities. The polytope in higher dimension, let's say Q, is the extension of the original polytope P and the minimum number of facets that Q can have is the extension complexity of P. Then the optimal solution of the problem over Q, gives the optimal solution over P. The natural question may raise is when is it possible to have an extension with a polynomial number of inequalities? Nondeterministic communication complexity is a powerful tool for proving lower bound on the extension complexity of a polytopes. Finding a suitable communication complexity problem corresponded to a polytope P and proving a linear lower bound for the nondeterministic communication complexity of it, will rule out all the attempts for finding sub-exponential size extension Q of P. In this thesis, we focus on the random Boolean functions f, with density p = p(n). We give tight upper and lower bounds for the nondeterministic communication complexity and parameters related to it. Also, we study the rank of fooling set matrix which is an important lower bound for nondeterministic communication complexity. Finally, we investigate the graph of the pedigree polytope. Pedigree polytope is an extension of TSP (traveling salesman problem; the most extensively studied problem in combinatorial optimization) polytopes with a nice combinatorial structure. The graph of a polytope can be regarded as an abstract graph and it reveals meaningful information about the properties of the polytope

On some problems related to 2-level polytopes

In this thesis we investigate a number of problems related to 2-level polytopes, in particular from the point of view of the combinatorial structure and the extension complexity. 2-level polytopes were introduced as a generalization of stable set polytopes of perfect graphs, and despite their apparently simple structure, are at the center of many open problems ranging from information theory to semidefinite programming. The extension complexity of a polytope P is a measure of the complexity of representing P: it is the smallest size of an extended formulation of P, which in turn is a linear description of a polyhedron that projects down to P. In the first chapter, we examine several classes of 2-level polytopes arising in combinatorial settings and we prove a relation between the number of vertices and facets of such polytopes, which is conjectured to hold for all 2-level polytopes. The proofs are obtained through an improved understanding of the combinatorial structure of such polytopes, which in some cases leads to results of independent interest. In the second chapter, we study the extension complexity of a restricted class of 2-level polytopes, the stable set polytopes of bipartite graphs, for which we obtain non-trivial lower and upper bounds. In the third chapter we study slack matrices of 2-level polytopes, important combinatorial objects related to extension complexity, defining operations on them and giving algorithms for the following recognition problem: given a matrix, determine whether it is a slack matrix of some special class of 2-level polytopes. In the fourth chapter we address the problem of explicitly obtaining small size extended formulations whose existence is guaranteed by communication protocols. In particular we give an algorithm to write down extended formulations for the stable set polytope of perfect graphs, making a well known result by Yannakakis constructive, and we extend this to all deterministic protocols

Interactions entre les Cliques et les Stables dans un Graphe

This thesis is concerned with different types of interactions between cliques and stable sets, two very important objects in graph theory, as well as with the connections between these interactions. At first, we study the classical problem of graph coloring, which can be stated in terms of partioning the vertices of the graph into stable sets. We present a coloring result for graphs with no triangle and no induced cycle of even length at least six. Secondly, we study the Erdös-Hajnal property, which asserts that the maximum size of a clique or a stable set is polynomial (instead of logarithmic in random graphs). We prove that the property holds for graphs with no induced path on k vertices and its complement.Then, we study the Clique-Stable Set Separation, which is a less known problem. The question is about the order of magnitude of the number of cuts needed to separate all the cliques from all the stable sets. This notion was introduced by Yannakakis when he studied extended formulations of the stable set polytope in perfect graphs. He proved that a quasi-polynomial number of cuts is always enough, and he asked if a polynomial number of cuts could suffice. Göös has just given a negative answer, but the question is open for restricted classes of graphs, in particular for perfect graphs. We prove that a polynomial number of cuts is enough for random graphs, and in several hereditary classes. To this end, some tools developed in the study of the Erdös-Hajnal property appear to be very helpful. We also establish the equivalence between the Clique-Stable set Separation problem and two other statements: the generalized Alon-Saks-Seymour conjecture and the Stubborn Problem, a Constraint Satisfaction Problem.Cette thèse s'intéresse à différents types d'interactions entre les cliques et les stables, deux objets très importants en théorie des graphes, ainsi qu'aux relations entre ces différentes interactions. En premier lieu, nous nous intéressons au problème classique de coloration de graphes, qui peut s'exprimer comme une partition des sommets du graphe en stables. Nous présentons un résultat de coloration pour les graphes sans triangles ni cycles pairs de longueur au moins 6. Dans un deuxième temps, nous prouvons la propriété d'Erdös-Hajnal, qui affirme que la taille maximale d'une clique ou d'un stable devient polynomiale (contre logarithmique dans les graphes aléatoires) dans le cas des graphes sans chemin induit à k sommets ni son complémentaire, quel que soit k.Enfin, un problème moins connu est la Clique-Stable séparation, où l'on cherche un ensemble de coupes permettant de séparer toute clique de tout stable. Cette notion a été introduite par Yannakakis lors de l’étude des formulations étendues du polytope des stables dans un graphe parfait. Il prouve qu’il existe toujours un séparateur Clique-Stable de taille quasi-polynomiale, et se demande si l'on peut se limiter à une taille polynomiale. Göös a récemment fourni une réponse négative, mais la question se pose encore pour des classes de graphes restreintes, en particulier pour les graphes parfaits. Nous prouvons une borne polynomiale pour la Clique-Stable séparation dans les graphes aléatoires et dans plusieurs classes héréditaires, en utilisant notamment des outils communs à l'étude de la conjecture d'Erdös-Hajnal. Nous décrivons également une équivalence entre la Clique-Stable séparation et deux autres problèmes  : la conjecture d'Alon-Saks-Seymour généralisée et le Problème Têtu, un problème de Satisfaction de Contraintes

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

Lower bounds on matrix factorization ranks via noncommutative polynomial optimization

We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the completely positive rank, and their symmetric analogues: the positive semidefinite rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples