Fooling sets and rank

An $n\times n$ matrix $M$ is called a \textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,\ell} M_{\ell,k} = 0$ for every $k\ne \ell$. Dietzfelbinger, Hromkovi{\v{c}}, and Schnitger (1996) showed that n \le (\mbox{rk} M)^2, regardless of over which field the rank is computed, and asked whether the exponent on \mbox{rk} M can be improved. We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size n = \binom{\mbox{rk} M+1}{2}. In nonzero characteristic, we construct an infinite family of matrices with n= (1+o(1))(\mbox{rk} M)^2.Comment: 10 pages. Now resolves the open problem also in characteristic

Communication Complexity with Defective Randomness

Starting with the two standard model of randomized communication complexity, we study the communication complexity of functions when the protocol has access to a defective source of randomness. Specifically, we consider both the public-randomness and private-randomness cases, while replacing the commonly postulated perfect randomness with distributions over ? bit strings that have min-entropy at least k ? ?. We present general upper and lower bounds on the communication complexity in these cases, where the bounds are typically linear in ?-k and also depend on the size of the fooling set for the function being computed and on its standard randomized complexity

On the limits of the communication complexity technique for proving lower bounds on the size of minimal NFA’s

AbstractIn contrast to the minimization of deterministic finite automata (DFA’s), the task of constructing a minimal nondeterministic finite automaton (NFA) for a given NFA is PSPACE-complete. Moreover, there are no polynomial approximation algorithms with a constant approximation ratio for estimating the number of states of minimal NFA’s.Since one is unable to efficiently estimate the size of a minimal NFA in an efficient way, one should ask at least for developing mathematical proof methods that help to prove good lower bounds on the size of a minimal NFA for a given regular language. Here we consider the robust and most successful lower bound proof technique that is based on communication complexity. In this paper it is proved that even a strong generalization of this method fails for some concrete regular languages.“To fail” is considered here in a very strong sense. There is an exponential gap between the size of a minimal NFA and the achievable lower bound for a specific sequence of regular languages.The generalization of the concept of communication protocols is also strong here. It is shown that cutting the input word into 2O(n1/4) pieces for a size n of a minimal nondeterministic finite automaton and investigating the necessary communication transfer between these pieces as parties of a multiparty protocol does not suffice to get good lower bounds on the size of minimal nondeterministic automata. It seems that for some regular languages one cannot really abstract from the automata model that cuts the input words into particular symbols of the alphabet and reads them one by one using its input head

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

Some results in communication complexity.

Mak, Yan Kei.Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (leaves 59-63).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.6Chapter 1.1 --- Historical background --- p.6Chapter 1.2 --- Why study communication complexity? --- p.7Chapter 1.3 --- Main ideas and results --- p.8Chapter 1.4 --- Recent development --- p.12Chapter 1.5 --- Structure of the thesis --- p.12Chapter 2 --- Deterministic Communication Complexity --- p.13Chapter 2.1 --- Definitions --- p.13Chapter 2.2 --- Tiling lower bound --- p.16Chapter 2.3 --- Fooling set lower bound --- p.21Chapter 2.4 --- Rank lower bound --- p.24Chapter 2.5 --- Comparison of the bounds --- p.27Chapter 3 --- Nondeterministic Communication Complexity --- p.29Chapter 3.1 --- Definitions --- p.29Chapter 3.2 --- "Gaps between N0(f), N1(f) and D(f)" --- p.31Chapter 3.3 --- Aho-Ullman-Yannakakis Theorem --- p.33Chapter 4 --- Randomized Communication Complexity --- p.38Chapter 4.1 --- Preliminaries --- p.38Chapter 4.2 --- Definitions --- p.39Chapter 4.3 --- Error reduction --- p.41Chapter 4.4 --- Exponential gap with D(f) --- p.42Chapter 4.5 --- The public coin model --- p.44Chapter 4.6 --- Distributional complexity --- p.46Chapter 5 --- Communication Complexity Classes --- p.51Chapter 5.1 --- Basic classes --- p.51Chapter 5.2 --- Polynomial-time hierarchy --- p.52Chapter 5.3 --- Reducibility and completeness --- p.53Chapter 6 --- Further topics --- p.56Chapter 6.1 --- Quantum communication complexity --- p.56Chapter 6.2 --- More techniques for bounds --- p.57Chapter 6.3 --- Complexity of communication complexity --- p.57Bibliography --- p.5

On limited interaction in communication complexity

Wir haben Interaktion in der Kommunikationskomplexität untersucht und dabei die drei Modi probabilistische, (beschränkt) nichtdeterministische und quantenmechanische Kommunikation betrachtet. Bei allen drei Modi haben wir herausgefunden, dass Interaktion für Effzienz oft unerlässlich ist, im nichtdeterministischen Fall gibt es eine Abhängigkeit zwischen dem Einfluss der Interaktion und der erlaubten Anzahl der nichtdeterministischen Ratebits. Abgesehen von dem erreichten besseren Verständnis des Kommunikationsmodells haben wir verschiedene Anwendungen auf andere Berechnungsmodelle beschrieben, bei denen untere Schranken der Kommunikation zu unteren Schranken für andere Ressourcen in diesen Modellen geführt haben. Ein Beispiel eines kommunikations- und interaktionsbeschränkten Modells sind endliche Automaten, welche wir in allen drei Modi untersucht haben. Ein weiteres Beispiel sind Formeln, für die wir eine Verbindung zwischen Einweg Kommunikation und Formellänge herstellen konnten. Diese Verbindung führte zu unteren Schranken für probabilistische, nichtdeterministische und Quanten Formeln. Dabei sind die unteren Schranken für Quanten Formeln und probabilistische Formeln im wesentlichen gleich. Für monotone Schaltkreise haben wir gezeigt, wie nichtdeterministisches Raten die Tiefe drastisch reduzieren kann, und wie eine geringfügige Einschränkung der nichtdeterministischen Ratebits zu einer Tiefenhierarchie führt. Insgesamt lässt sich feststellen, dass die Schwäche interaktionsbeschränkter Kommunikation mathematisch nachvollziehbar ist. Außerdem scheint ein solches Verhalten in der Welt einfacher Berechnungsmodelle häufig aufzutreten. Oder anders gesagt, viele Berechnungsmodelle sind deshalb einfacher zu verstehen, weil sie durch interaktionsbeschränkte Kommunikation analysierbar sind

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