85,247 research outputs found
Dynamic concentration of the triangle-free process
The triangle-free process begins with an empty graph on n vertices and
iteratively adds edges chosen uniformly at random subject to the constraint
that no triangle is formed. We determine the asymptotic number of edges in the
maximal triangle-free graph at which the triangle-free process terminates. We
also bound the independence number of this graph, which gives an improved lower
bound on the Ramsey numbers R(3,t): we show R(3,t) > (1-o(1)) t^2 / (4 log t),
which is within a 4+o(1) factor of the best known upper bound. Our improvement
on previous analyses of this process exploits the self-correcting nature of key
statistics of the process. Furthermore, we determine which bounded size
subgraphs are likely to appear in the maximal triangle-free graph produced by
the triangle-free process: they are precisely those triangle-free graphs with
density at most 2.Comment: 75 pages, 1 figur
On the Minimax Regret for Online Learning with Feedback Graphs
In this work, we improve on the upper and lower bounds for the regret of
online learning with strongly observable undirected feedback graphs. The best
known upper bound for this problem is , where is the number of actions, is the independence
number of the graph, and is the time horizon. The factor is
known to be necessary when (the experts case). On the other hand,
when (the bandits case), the minimax rate is known to be
, and a lower bound is known to hold for any . Our improved upper bound
holds for any
and matches the lower bounds for bandits and experts, while
interpolating intermediate cases. To prove this result, we use FTRL with
-Tsallis entropy for a carefully chosen value of that
varies with . The analysis of this algorithm requires a new bound on
the variance term in the regret. We also show how to extend our techniques to
time-varying graphs, without requiring prior knowledge of their independence
numbers. Our upper bound is complemented by an improved
lower bound for all
, whose analysis relies on a novel reduction to multitask learning.
This shows that a logarithmic factor is necessary as soon as
Independent sets and cuts in large-girth regular graphs
We present a local algorithm producing an independent set of expected size
on large-girth 3-regular graphs and on large-girth
4-regular graphs. We also construct a cut (or bisection or bipartite subgraph)
with edges on large-girth 3-regular graphs. These decrease the gaps
between the best known upper and lower bounds from to , from
to and from to , respectively. We are using
local algorithms, therefore, the method also provides upper bounds for the
fractional coloring numbers of and and fractional edge coloring number . Our algorithms are applications of the technique introduced by Hoppen
and Wormald
Counting independent sets in triangle-free graphs
Ajtai, Koml\'os, and Szemer\'edi proved that for sufficiently large every
triangle-free graph with vertices and average degree has an independent
set of size at least . We extend this by proving that
the number of independent sets in such a graph is at least This result is sharp for infinitely many
apart from the constant. An easy consequence of our result is that there
exists such that every -vertex triangle-free graph has at least independent sets. We conjecture that the exponent above
can be improved to . This would be sharp by the
celebrated result of Kim which shows that the Ramsey number has order
of magnitude
New lower bounds for the independence number of sparse graphs and hypergraphs
We obtain new lower bounds for the independence number of -free graphs
and linear -uniform hypergraphs in terms of the degree sequence. This
answers some old questions raised by Caro and Tuza \cite{CT91}. Our proof
technique is an extension of a method of Caro and Wei \cite{CA79, WE79}, and we
also give a new short proof of the main result of \cite{CT91} using this
approach. As byproducts, we also obtain some non-trivial identities involving
binomial coefficients
Density version of the Ramsey problem and the directed Ramsey problem
We discuss a variant of the Ramsey and the directed Ramsey problem. First,
consider a complete graph on vertices and a two-coloring of the edges such
that every edge is colored with at least one color and the number of bicolored
edges is given. The aim is to find the maximal size of a
monochromatic clique which is guaranteed by such a coloring. Analogously, in
the second problem we consider semicomplete digraph on vertices such that
the number of bi-oriented edges is given. The aim is to bound the
size of the maximal transitive subtournament that is guaranteed by such a
digraph.
Applying probabilistic and analytic tools and constructive methods we show
that if , (), then where only depend on , while if then . The latter case is
strongly connected to Tur\'an-type extremal graph theory.Comment: 17 pages. Further lower bound added in case $|E_{RB}|=|E_{bi}| =
p{n\choose 2}
A Quantum Lovasz Local Lemma
The Lovasz Local Lemma (LLL) is a powerful tool in probability theory to show
the existence of combinatorial objects meeting a prescribed collection of
"weakly dependent" criteria. We show that the LLL extends to a much more
general geometric setting, where events are replaced with subspaces and
probability is replaced with relative dimension, which allows to lower bound
the dimension of the intersection of vector spaces under certain independence
conditions. Our result immediately applies to the k-QSAT problem: For instance
we show that any collection of rank 1 projectors with the property that each
qubit appears in at most of them, has a joint satisfiable
state.
We then apply our results to the recently studied model of random k-QSAT.
Recent works have shown that the satisfiable region extends up to a density of
1 in the large k limit, where the density is the ratio of projectors to qubits.
Using a hybrid approach building on work by Laumann et al. we greatly extend
the known satisfiable region for random k-QSAT to a density of
. Since our tool allows us to show the existence of joint
satisfying states without the need to construct them, we are able to penetrate
into regions where the satisfying states are conjectured to be entangled,
avoiding the need to construct them, which has limited previous approaches to
product states.Comment: 19 page
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