983 research outputs found
Tur\'annical hypergraphs
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
A sparsifier of a graph (Bencz\'ur and Karger; Spielman and Teng) is a
sparse weighted subgraph that approximately retains the cut
structure of . 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 is not approximated within a
multiplicative factor , but is, instead, approximated up to an
additive term bounded by times , where
is the average degree, and is the sum of the degrees of the
vertices in . We provide a probabilistic polynomial time construction of
such sparsifiers for every graph, and our sparsifiers have a near-optimal
number of edges . We also provide
a deterministic polynomial time construction that constructs sparsifiers with a
weaker property having the optimal number of edges . Our
constructions also satisfy a spectral version of the ``additive
sparsification'' property.
Our construction of ``additive sparsifiers'' with edges also
works for hypergraphs, and provides the first non-trivial notion of
sparsification for hypergraphs achievable with hyperedges when
and the rank of the hyperedges are constant. Finally, we provide
a new construction of spectral hypergraph sparsifiers, according to the
standard definition, with
hyperedges, improving over the previous spectral construction (Soma and
Yoshida) that used hyperedges even for constant and
.Comment: 31 page
Fast Local Computation Algorithms
For input , let denote the set of outputs that are the "legal"
answers for a computational problem . Suppose and members of 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 ,
support queries by a user to values of specified locations in a legal
output . When more than one legal output exists for a given
, the local computation algorithm should output in a way that is consistent
with at least one such . 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 -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 -SAT formulas.Comment: A preliminary version of this paper appeared in ICS 2011, pp. 223-23
Differential equation approximations for Markov chains
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
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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