Random @math -intersection graphs have recently received much interest in a wide range of application areas. Broadly speaking, a random @math -intersection graph is constructed by first assigning each vertex a set of items in some random manner, and then putting an undirected edge between all pairs of vertices that share at least @math items (the graph is called a random intersection graph when @math ). A special case of particular interest is a uniform random @math -intersection graph, where each vertex independently selects the same number of items uniformly at random from a common item pool. Another important case is a binomial random @math -intersection graph, where each item from a pool is independently assigned to each vertex with the same probability. Both models have found numerous applications thus far including cryptanalysis, and the modeling of recommender systems, secure sensor networks, online social networks, trust networks and small-world networks (uniform random @math -intersection graphs), as well as clustering analysis, classification, and the design of integrated circuits (binomial random @math -intersection graphs). In this paper, for binomial uniform random @math -intersection graphs, we present results related to @math -connectivity and minimum vertex degree. Specifically, we derive the asymptotically exact probabilities and zero-one laws for the following three properties: (i) @math -vertex-connectivity, (ii) @math -edge-connectivity and (iii) the property of minimum vertex degree being at least @math .
& 3 * -1.5pt @math -1.5pt & 2 * & 2 * -1.5pt exact probabilities -1.5pt & -1.5pt 5-5 & & & & 3-5 & & & -1.5pt exact probabilities & 2-5 & 4 * -1.5pt @math -1.5pt & 2 * & -1.5pt exact probabilities -1.5pt & -1.5pt @cite_22 4-5 & & & -1.5pt zero--one laws & -1.5pt @cite_35 @cite_5 3-5 & & 2 * & -1.5pt exact probabilities -1.5pt & -1.5pt @cite_18 4-5 & & & -1.5pt zero--one laws & -1.5pt @cite_16 @cite_29 8 * & 4 * -1.5pt @math -1.5pt & 2 * & -1.5pt exact probabilities -1.5pt & 4 * -1.5pt -1.5pt 4-4 & & & -1.5pt zero--one laws & 3-4 & & 2 * & -1.5pt exact probabilities -1.5pt & 4-4 & & & -1.5pt zero--one laws & 2-5 & 4 * -1.5pt @math -1.5pt & 2 * & -1.5pt exact probabilities -1.5pt & -1.5pt @cite_22 4-5 & & & -1.5pt zero--one laws -1.5pt & -1.5pt @cite_35 3-5 & & 2 * & -1.5pt exact probabilities -1.5pt & -1.5pt @cite_10 4-5 & & & -1.5pt zero--one laws & -1.5pt @cite_8 @cite_39 2.5
@cite_35 @cite_18 @cite_22 @cite_8 @cite_29 @cite_39 @cite_5 @cite_16 @cite_10
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Pour des echantillons aleatoires independants: X 1 X 2 …X n1 distribue selon f(x) et Y 1 Y 2 …Y n2 distribue selon g(x) on presente un test qui possede les proprietes suivantes: la probabilite d'erreur de premiere espece ne depend pas de f; lorsque min (n 1 , n 2 )→∞, la statistique de test est asymptotiquement independante de la distribution sous Ho et la distribution limite est connue; le test est consistant contre des alternatives generales Soit Ω un domaine borne dans R n , n≥3 avec une frontiere C 1,1 . On considere l'operateur Lγ(u)=⊇•(γ⊇u) ou γ(x) est une fonction a valeur reelle dans C 1,1 (Ω) avec une borne superieure positive. On definit la forme quadratique Qγ sur H 1 2 (∂Ω) par Qγ(f)=∫ Ω γ(x)|⊇u(x)| 2 dx ou u∈H 1 (Ω) est la solution unique a Lγu=0 dans Ω, u| ∂Ω =f. On etudie la reconstruction de γ a partir de Qγ Considerons une marche aleatoire branchante surcritique a temps discret. Nous nous interessons a la probabilite qu'il existe un rayon infini du support de la marche aleatoire branchante, le long duquel elle croit plus vite qu'une fonction lineaire de pente y - e, ou γ designe la vitesse asymptotique de la position de la particule la plus a droite dans la marche aleatoire branchante. Sous des hypotheses generales peu restrictives, nous prouvons que, lorsque e → 0, cette probabilite decroit comme exp —β+o(1) e 1 2 , ou β est une constante strictement positive dont la valeur depend de la loi de la marche aleatoire branchante. Dans le cas special ou des variables aleatoires i.i.d. de Bernoulli(p) (avec 0 < p < 1 2 ) sont placees sur les aretes d'un arbre binaire enracine, ceci repond a une question ouverte de Robin Pemantle (Ann. Appl. Probab. 19 (2009) 1273-1291). On presente un algorithme polynomial permettant de determiner un sous-graphe biparti d'un graphe G sans triangle ni boucle de degre maximum 3, contenant au moins 4 5 des aretes de G. On caracterise le dodecaedre et le graphe de Petersen comme les seuls graphes connexes 3-reguliers sans triangle ni boucle pour lesquels il n'existe pas de sous graphe biparti ayant un nombre d'aretes superieur a cette proportion 3 Voting 17 3.1 Voting Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.1 Scoring Rules . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.2 Condorcet Extensions . . . . . . . . . . . . . . . . . . . 19 3.1.3 Other Rules . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.1 The Gibbard-Satterthwaite Impossibility . . . . . . . . . . 24 3.2.2 Restricted Domains of Preferences . . . . . . . . . . . . . 24 3.2.3 Computational Hardness of Manipulation . . . . . . . . . 26 3.2.4 Probabilistic Voting Rules . . . . . . . . . . . . . . . . . 29 3.2.5 Irresolute Voting Rules . . . . . . . . . . . . . . . . . . . 30 3.3 Possible and Necessary Winners . . . . . . . . . . . . . . . . . . 31 Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP ⊆ naPCP1,s[O(log n),3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems ("3-CSPs") is NP-hard. The previous best upper bound and lower bound for s are 20 27+µ by Khot and Saket [KS06], and 5 8 (assuming NP subseteq BPP) by Zwick [Zwi98]. In this paper we close the gap assuming Khot's d-to-1 Conjecture. Formally, we prove that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5 8+ µ[O(log n),3] for any constant µ > 0. Our conditional result also solves Hastad's open question [Has01] on determining the inapproximability of satisfiable Max-NTW ("Not Two") instances and confirms Zwick's conjecture [Zwi98] that the 5 8-approximation algorithm for satisfiable 3-CSPs is optimal. Dans cette Note on determine deux fonctions x et y, uniformes et continues d’une variable (reelle) t, qui, lorsque t varie dans l’intervalle (0, 1), prennent toutes les couples de valeurs telles que 0≤x≤1, 0≤y≤1. Si l’on appelle, suivant l’usage, courbe continue le lieu des points dont les coordonnees sont des fonctions continues d’une variable, on a ainsi un arc de courbe qui passe par tous les points d’un carre. Donc, etant donne un arc de courbe continue, sans faire d’autres hypotheses, il n’est pas toujours possible de le renfermer dans une aire arbitrairement petite. Les principes de l'ethique biomedicale constituent, par leur influence, l’ouvrage majeur de l’ethique medicale contemporaine. Au-dela de son contexte nord-americain d’elaboration, de la deontologie medicale traditionnelle et des theories morales classiques, la reflexion proposee a reconfigure l’analyse des questions ethiques liees a la relation de soin et au monde de la sante. Depuis la premiere edition de ce texte (1979), les auteurs n’ont eu de cesse de le remanier et d’en presenter des versions integrant toujours davantage leurs reponses aux objections et critiques qu’ils recevaient ou se formulaient eux-memes. C’est la 5e edition (dans l’attente de la 6e), datant de 2001, qui est ici traduite pour la premiere fois. Livre en debat, ne lui-meme des discussions engagees aux Etats-Unis dans les annees 1970, notamment sur l’ethique des essais cliniques, il vise a determiner des principes-reperes susceptibles d’eclairer les pratiques medicales et les argumentations qui les accompagnent des lors qu’elles engagent un rapport aux normes et aux valeurs : le principe d’autonomie, le principe de non-malfaisance, le principe de bienfaisance et le principe de justice. Le statut de la theorie dans la vie morale, les rapports entre les principes et les elements factuels, les rapports des differents principes entre eux sont les problemes qui structurent continument la reflexion des auteurs. Ainsi s’edifie une approche se voulant fine, souple et rigoureuse de ce qui peut justifier et guider le raisonnement ethique dans les prestations de sante et les relations de soin. 1. Summary In Part I, four ostensibly different theoretical models of induction are presented, in which the problem dealt with is the extrapolation of a very long sequence of symbols—presumably containing all of the information to be used in the induction. Almost all, if not all problems in induction can be put in this form. Some strong heuristic arguments have been obtained for the equivalence of the last three models. One of these models is equivalent to a Bayes formulation, in which a priori probabilities are assigned to sequences of symbols on the basis of the lengths of inputs to a universal Turing machine that are required to produce the sequence of interest as output. Though it seems likely, it is not certain whether the first of the four models is equivalent to the other three. Few rigorous results are presented. Informal investigations are made of the properties of these models. There are discussions of their consistency and meaningfulness, of their degree of independence of the exact nature of the Turing machine used, and of the accuracy of their predictions in comparison to those of other induction methods. In Part II these models are applied to the solution of three problems—prediction of the Bernoulli sequence, extrapolation of a certain kind of Markov chain, and the use of phrase structure grammars for induction. Though some approximations are used, the first of these problems is treated most rigorously. The result is Laplace's rule of succession. The solution to the second problem uses less certain approximations, but the properties of the solution that are discussed, are fairly independent of these approximations. The third application, using phrase structure grammars, is least exact of the three. First a formal solution is presented. Though it appears to have certain deficiencies, it is hoped that presentation of this admittedly inadequate model will suggest acceptable improvements in it. This formal solution is then applied in an approximate way to the determination of the “optimum” phrase structure grammar for a given set of strings. The results that are obtained are plausible, but subject to the uncertainties of the approximation used. Presentation d'une methode de type geometrique pour denombrer les arbres binaires de taille n; on s'interesse d'abord a un type d'objets plus complexes les arbres binaires a feuilles numerotees, et on obtient pour ceux-ci un procede de construction iteratif tres simple, l'insertion des feuilles par numeros croissants In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30], several inequalities for tail probabilities of sums M n = X 1 + ... + X n of bounded independent random variables X j were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Etudes Sci. Publ. Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257-314] and refined (and extended) by me. In this article, I introduce a new type of inequality. Namely, I show that P M n ≥ x ≤ cP S n ≥ x , where c is an absolute constant and S n = e 1 + ... + e n is a sum of independent identically distributed Bernoulli random variables (a random variable is called Bernoulli if it assumes at most two values). The inequality holds for those x ∈ Ρ where the survival function x → P S n ≥ x has a jump down. For the remaining x the inequality still holds provided that the function between the adjacent jump points is interpolated linearly or log-linearly. If it is necessary, to estimate P S n ≥ x special bounds can be used for binomial probabilities. The results extend to martingales with bounded differences. It is apparent that Theorem 1.1 of this article is the most important. The inequalities have applications to measure concentration, leading to results of the type where, up to an absolute constant, the measure concentration is dominated by the concentration in a simplest appropriate model, such results will be considered elsewhere. It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error @math in trace distance required @math copies for a @math -dimensional density matrix of rank @math . Here, we give a theoretical measurement scheme (POVM) that requires @math copies to estimate @math to error @math in infidelity, and a matching lower bound up to logarithmic factors. This implies @math copies suffice to achieve error @math in trace distance. We also prove that for independent (product) measurements, @math copies are necessary in order to achieve error @math in infidelity. For fixed @math , our measurement can be implemented on a quantum computer in time polynomial in @math . Let us denote by R(k, ⩾ λ)[R(k, ⩽ λ)] the maximal number M such that there exist M different permutations of the set 1,…, k such that any two of them have at least λ (at most λ, respectively) common positions. We prove the inequalities R(k, ⩽ λ) ⩽ kR(k − 1, ⩽ λ − 1), R(k, ⩾ λ) ⩾ R(k, ⩽ λ − 1) ⩽ k!, R(k, ⩾ λ) ⩽ kR(k − 1, ⩾ λ − 1). We show: R(k, ⩾ k − 2) = 2, R(k, ⩾ 1) = (k − 1)!, R(pm, ⩾ 2) = (pm − 2)!, R(pm + 1, ⩾ 3) = (pm − 2)!, R(k, ⩽ k − 3) = k!2, R(k, ⩽ 0) = k, R(pm, ⩽ 1) = pm(pm − 1), R(pm + 1, ⩽ 2) = (pm + 1)pm(pm − 1). The exact value of R(k, ⩾ λ) is determined whenever k ⩾ k0(k − λ); we conjecture that R(k, ⩾ λ) = (k − λ)! for k ⩾ k0(λ). Bounds for the general case are given and are used to determine that the minimum of |R(k, ⩾ λ) − R(k, ⩽ λ)| is attained for λ = (k2) + O(klog k). par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . We consider the problem of approximating a Boolean functionf∶ 0,1 n → 0,1 by the sign of an integer polynomialp of degreek. For us, a polynomialp(x) predicts the value off(x) if, wheneverp(x)≥0,f(x)=1, and wheneverp(x)<0,f(x)=0. A low-degree polynomialp is a good approximator forf if it predictsf at almost all points. Given a positive integerk, and a Boolean functionf, we ask, “how good is the best degreek approximation tof?” We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the casef is parity. Minsky and Papert [10] proved that a perceptron cannot compute parity; our bounds indicate exactly how well a perceptron canapproximate it. As a consequence, we are able to give the first correct proof that, for a random oracleA, PP A is properly contained in PSPACE A . We are also able to prove the old AC0 exponential-size lower bounds in a new way. This allows us to prove the new result that an AC0 circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials. A new coding problem is introduced for a correlated source (X sup n ,Y sup n ) sub n=1 sup spl infin . The observer of X sup n can transmit data depending on X sup n at a prescribed rate R. Based on these data the observer of Y sup n tries to identify whether for some distortion measure spl rho (like the Hamming distance) n sup -1 spl rho (X sup n ,Y sup n ) spl les d, a prescribed fidelity criterion. We investigate as functions of R and d the exponents of two error probabilities, the probabilities for misacceptance, and the probabilities for misrejection. In the case where X sup n and Y sup n are independent, we completely characterize the achievable region for the rate R and the exponents of two error probabilities; in the case where X sup n and Y sup n are correlated, we get some interesting partial results for the achievable region. During the process, we develop a new method for proving converses, which is called "the inherently typical subset lemma". This new method goes considerably beyond the "entropy characterization" the "image size characterization," and its extensions. It is conceivable that this new method has a strong impact on multiuser information theory. Abstract At the heart of this article will be the study of a branching Brownian motion (BBM) with killing , where individual particles move as Brownian motions with drift − ρ , perform dyadic branching at rate β and are killed on hitting the origin. Firstly, by considering properties of the right-most particle and the extinction probability, we will provide a probabilistic proof of the classical result that the ‘one-sided’ FKPP travelling-wave equation of speed − ρ with solutions f : [ 0 , ∞ ) → [ 0 , 1 ] satisfying f ( 0 ) = 1 and f ( ∞ ) = 0 has a unique solution with a particular asymptotic when ρ 2 β , and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of [S.C. Harris, Travelling-waves for the FKPP equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A 129 (3) (1999) 503–517] and [A.E. Kyprianou, Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis, Ann. Inst. H. Poincare Probab. Statist. 40 (1) (2004) 53–72] and includes an intuitive application of a change of measure inducing a spine decomposition that, as a by product, gives the new result that the asymptotic speed of the right-most particle in the killed BBM is 2 β − ρ on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, W λ , that appears of fundamental importance as well as facilitating some new results on the almost-sure exponential growth rate of the number of particles of speed λ ∈ ( 0 , 2 β − ρ ) . Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed λ > 2 β − ρ . This result combined with Chauvin and Rouault's [B. Chauvin, A. Rouault, KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Related Fields 80 (2) (1988) 299–314] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals W λ as the limiting Radon–Nikodým derivative when conditioning the right-most particle to travel at speed λ into the distant future. On considere la classe de toutes les densites unimodales definies sur un intervalle de longueur L et borne par H; on etudie le risque minimax sur cette classe, lorsqu'on estime en utilisant n observations i.i.d., la perte etant mesuree par la distance L 1 entre l'estimateur et la densite reelle Let (M t )0≤t≤1 be any martingale with initial law M0 μ0 and terminal law M1 μ1 and let S≡sup0≤t≤1M t . Then there is an upper bound, with respect to stochastic ordering of probability measures, on the law of S.