Large deviations for sums of partly dependent random variables

Janson, Svante

doi:10.1002/rsa.20008

Cited by 147 publications

(203 citation statements)

References 20 publications

(32 reference statements)

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“…In conclusion, the combination of (6) and (8) implies that, for large n, algorithm A outputs a nearly balanced recut on Π (G, * ) with high probability. By the discussion in Sect.…”

Section: Local Concentration Boundmentioning

confidence: 86%

“…Since the maximum degree of G 2r is at most max v |B G (v, 2r)| = o(n), this graph can always be partitioned into χ(G 2r ) = o(n) independent sets. Indeed, Janson [8] presents large deviation bounds for sums of type (7) by applying Chernoff-Hoeffding bounds for each colour class in a χ(G 2r )-colouring of G 2r . For any ε > 0, Theorem 2.1 in Janson [8], as applied to our setting, gives…”

Section: Local Concentration Boundmentioning

confidence: 99%

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No sublogarithmic-time approximation scheme for bipartite vertex cover

Göös

Suomela

2013

Distrib. Comput.

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König's theorem states that on bipartite graphs the size of a maximum matching equals the size of a minimum vertex cover. It is known from prior work that for every ε > 0 there exists a constant-time distributed algorithm that finds a (1 + ε)-approximation of a maximum matching on bounded-degree graphs. In this work, we show-somewhat surprisingly-that no sublogarithmic-time approximation scheme exists for the dual problem: there is a constant δ > 0 so that no randomised distributed algorithm with running time o(log n) can find a (1 + δ )-approximation of a minimum vertex cover on 2-coloured graphs of maximum degree 3. In fact, a simple application of the Linial-Saks (1993) decomposition demonstrates that this run-time lower bound is tight.Our lower-bound construction is simple and, to some extent, independent of previous techniques. Along the way we prove that a certain cut minimisation problem, which might be of independent interest, is hard to approximate locally on expander graphs.

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“…In conclusion, the combination of (6) and (8) implies that, for large n, algorithm A outputs a nearly balanced recut on Π (G, * ) with high probability. By the discussion in Sect.…”

Section: Local Concentration Boundmentioning

confidence: 86%

Section: Local Concentration Boundmentioning

confidence: 99%

No sublogarithmic-time approximation scheme for bipartite vertex cover

Göös

Suomela

2013

Distrib. Comput.

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“…Applications of the Hajnal-Szemerédi theorem and recent results on equitable colorings of graphs can be found in (among others) [1], [2], [9], [11], [12], [19]. Equitable coloring turned out to be useful in establishing bounds on tails of sums of dependent variables [6], [8], [18].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Equitable coloring of random graphs

Krivelevich

Patkós²

2009

Random Struct Algorithms

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“…Let X follow a binomial distribution with parameters n and p. Then and P(X np + t) exp − t Proofs of these estimates may be found in [25]; see also [3] and [1]. They can be derived from an application of Markov's inequality to the random variable e λ(X−np) using the fact that the moment generating function Ee λ(X−np) is e −λpn (pe λ + 1 − p) n .…”

Section: 2mentioning

confidence: 99%

A Probabilistic Technique for Finding Almost-Periods of Convolutions

2010

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Abstract. We introduce a new probabilistic technique for finding 'almost-periods' of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth's theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain-Green theorem on the existence of long arithmetic progressions in sumsets A + B that works with sparser subsets of {1, . . . , N } than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additive combinatorics, showing that product sets A 1 · A 2 · A 3 and A 2 · A −2 are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in A · B.Our results are 'local' in nature, meaning that it is not necessary for the sets under consideration to be dense in the ambient group. In particular, our results apply to finite subsets of infinite groups provided they 'interact nicely' with some other set.

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Large deviations for sums of partly dependent random variables

Cited by 147 publications

References 20 publications

No sublogarithmic-time approximation scheme for bipartite vertex cover

No sublogarithmic-time approximation scheme for bipartite vertex cover

Equitable coloring of random graphs

A Probabilistic Technique for Finding Almost-Periods of Convolutions

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