We consider a random graph on a given degree sequence @math , satisfying certain conditions. We focus on two parameters @math . Molloy and Reed proved that Q=0 is the threshold for the random graph to have a giant component. We prove that if @math then, with high probability, the size of the largest component of the random graph will be of order @math . If @math is asymptotically larger than @math then the size of the largest component is asymptotically smaller or larger than @math . Thus, we establish that the scaling window is @math .
Kang and Seierstad @cite_4 applied generating functions to study the case where @math , but is outside of the scaling window. They require a maximum degree of at most @math and that the degree sequences satisfy certain conditions that are stronger than those in @cite_12 ; one of these conditions implies that @math is bounded by a constant. Based on what is known for @math , it was natural to guess that for @math we would have @math . They proved that if @math then @math , and if @math then @math . So for the case where @math is bounded, this almost confirmed that natural guess - except that they did not cover the range where @math .
@cite_4 @cite_12
2001663593 2950427777 2146489966 2949393806
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 characterize the set of properties of Boolean-valued functions on a finite domain @math that are testable with a constant number of samples. Specifically, we show that a property @math is testable with a constant number of samples if and only if it is (essentially) a @math -part symmetric property for some constant @math , where a property is @math -part symmetric if there is a partition @math of @math such that whether @math satisfies the property is determined solely by the densities of @math on @math . We use this characterization to obtain a number of corollaries, namely: (i) A graph property @math is testable with a constant number of samples if and only if whether a graph @math satisfies @math is (essentially) determined by the edge density of @math . (ii) An affine-invariant property @math of functions @math is testable with a constant number of samples if and only if whether @math satisfies @math is (essentially) determined by the density of @math . (iii) For every constant @math , monotonicity of functions @math on the @math -dimensional hypergrid is testable with a constant number of samples. We study compositions of the integer @math in which the first part, successive differences, and the last part are constrained to lie in prescribed sets @math , respectively. A simple condition on @math guarantees that the generating function @math has only a simple pole on its circle of convergence and this at @math , a function independent of @math and @math . Thus the number of compositions is asymptotic to @math for a suitable constant @math . We prove a multivariate central and local limit theorem and apply it to various statistics of random locally restricted compositions of @math , such as number of parts, numbers of parts of given sizes, and number of rises. The first and last parts are shown to have limiting distributions and to be asymptotically independent. If @math has only finitely many positive elements @math , or finitely many negative elements @math , then the largest part and number of distinct part sizes are almost surely @math . On the other hand, when both @math and @math have a positive asymptotic lower "local log-density", we prove that the largest part and number of distinct part sizes are almost surely @math , and we give sufficient conditions for the largest part to be almost surely asymptotic to @math . We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k ( ) ( ) ;, where k is the alphabet size of I, @math , and @math denotes the number of eigenvalues larger than @math in the normalized adjacency matrix of the constraint graph of @math . In the case that @math is a instance, the threshold @math is only a polynomial in @math , and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the @math constraints specified by the @math level of the Lasserre hierarchy, and in some cases @math rounds of our program can be evaluated in time @math .