983 research outputs found

    Tur\'annical hypergraphs

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    This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turan's theorem, which deals with graphs G=([n],E) such that no member of the restriction set consisting of all r-tuples on [n] induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced just by all r-tuples touching a given m-element set. That is, we determine the maximal number of edges in an n-vertex such that no K_r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r-uniform hypergraph R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if for any graph G on [n] with more edges than the Turan number ex(n,K_r) (respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G. We determine the thresholds for random r-uniform hypergraphs to be Turannical and to epsilon-Turannical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht [Extremal results for random discrete structures] to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random graphs.Comment: 33 pages, minor improvements thanks to two referee

    New Notions and Constructions of Sparsification for Graphs and Hypergraphs

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    A sparsifier of a graph GG (Bencz\'ur and Karger; Spielman and Teng) is a sparse weighted subgraph G~\tilde G that approximately retains the cut structure of GG. For general graphs, non-trivial sparsification is possible only by using weighted graphs in which different edges have different weights. Even for graphs that admit unweighted sparsifiers, there are no known polynomial time algorithms that find such unweighted sparsifiers. We study a weaker notion of sparsification suggested by Oveis Gharan, in which the number of edges in each cut (S,Sˉ)(S,\bar S) is not approximated within a multiplicative factor (1+ϵ)(1+\epsilon), but is, instead, approximated up to an additive term bounded by ϵ\epsilon times dS+vol(S)d\cdot |S| + \text{vol}(S), where dd is the average degree, and vol(S)\text{vol}(S) is the sum of the degrees of the vertices in SS. We provide a probabilistic polynomial time construction of such sparsifiers for every graph, and our sparsifiers have a near-optimal number of edges O(ϵ2npolylog(1/ϵ))O(\epsilon^{-2} n {\rm polylog}(1/\epsilon)). We also provide a deterministic polynomial time construction that constructs sparsifiers with a weaker property having the optimal number of edges O(ϵ2n)O(\epsilon^{-2} n). Our constructions also satisfy a spectral version of the ``additive sparsification'' property. Our construction of ``additive sparsifiers'' with Oϵ(n)O_\epsilon (n) edges also works for hypergraphs, and provides the first non-trivial notion of sparsification for hypergraphs achievable with O(n)O(n) hyperedges when ϵ\epsilon and the rank rr of the hyperedges are constant. Finally, we provide a new construction of spectral hypergraph sparsifiers, according to the standard definition, with poly(ϵ1,r)nlogn{\rm poly}(\epsilon^{-1},r)\cdot n\log n hyperedges, improving over the previous spectral construction (Soma and Yoshida) that used O~(n3)\tilde O(n^3) hyperedges even for constant rr and ϵ\epsilon.Comment: 31 page

    Fast Local Computation Algorithms

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    For input xx, let F(x)F(x) denote the set of outputs that are the "legal" answers for a computational problem FF. Suppose xx and members of F(x)F(x) are so large that there is not time to read them in their entirety. We propose a model of {\em local computation algorithms} which for a given input xx, support queries by a user to values of specified locations yiy_i in a legal output yF(x)y \in F(x). When more than one legal output yy exists for a given xx, the local computation algorithm should output in a way that is consistent with at least one such yy. Local computation algorithms are intended to distill the common features of several concepts that have appeared in various algorithmic subfields, including local distributed computation, local algorithms, locally decodable codes, and local reconstruction. We develop a technique, based on known constructions of small sample spaces of kk-wise independent random variables and Beck's analysis in his algorithmic approach to the Lov{\'{a}}sz Local Lemma, which under certain conditions can be applied to construct local computation algorithms that run in {\em polylogarithmic} time and space. We apply this technique to maximal independent set computations, scheduling radio network broadcasts, hypergraph coloring and satisfying kk-SAT formulas.Comment: A preliminary version of this paper appeared in ICS 2011, pp. 223-23

    Differential equation approximations for Markov chains

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    We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs.Comment: Published in at http://dx.doi.org/10.1214/07-PS121 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Tight Thresholds for Cuckoo Hashing via XORSAT

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    We settle the question of tight thresholds for offline cuckoo hashing. The problem can be stated as follows: we have n keys to be hashed into m buckets each capable of holding a single key. Each key has k >= 3 (distinct) associated buckets chosen uniformly at random and independently of the choices of other keys. A hash table can be constructed successfully if each key can be placed into one of its buckets. We seek thresholds alpha_k such that, as n goes to infinity, if n/m <= alpha for some alpha < alpha_k then a hash table can be constructed successfully with high probability, and if n/m >= alpha for some alpha > alpha_k a hash table cannot be constructed successfully with high probability. Here we are considering the offline version of the problem, where all keys and hash values are given, so the problem is equivalent to previous models of multiple-choice hashing. We find the thresholds for all values of k > 2 by showing that they are in fact the same as the previously known thresholds for the random k-XORSAT problem. We then extend these results to the setting where keys can have differing number of choices, and provide evidence in the form of an algorithm for a conjecture extending this result to cuckoo hash tables that store multiple keys in a bucket.Comment: Revision 3 contains missing details of proofs, as appendix
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