pinecone/core-2020-05-10-deduplication huggingface dataset:
Coloring random graphs online without creating monochromatic subgraphs

Consider the following random process: The vertices of a binomial random
graph $G_{n,p}$ are revealed one by one, and at each step only the edges
induced by the already revealed vertices are visible. Our goal is to assign to
each vertex one from a fixed number $r$ of available colors immediately and
irrevocably without creating a monochromatic copy of some fixed graph $F$ in
the process. Our first main result is that for any $F$ and $r$, the threshold
function for this problem is given by $p_0(F,r,n)=n^{-1/m_1^*(F,r)}$, where
$m_1^*(F,r)$ denotes the so-called \emph{online vertex-Ramsey density} of $F$
and $r$. This parameter is defined via a purely deterministic two-player game,
in which the random process is replaced by an adversary that is subject to
certain restrictions inherited from the random setting. Our second main result
states that for any $F$ and $r$, the online vertex-Ramsey density $m_1^*(F,r)$
is a computable rational number. Our lower bound proof is algorithmic, i.e., we
obtain polynomial-time online algorithms that succeed in coloring $G_{n,p}$ as
desired with probability $1-o(1)$ for any $p(n) = o(n^{-1/m_1^*(F,r)})$.Comment: some minor addition
Upper bounds on probability thresholds for asymmetric Ramsey properties

Given two graphs $G$ and $H$, we investigate for which functions $p=p(n)$ the
random graph $G_{n,p}$ (the binomial random graph on $n$ vertices with edge
probability $p$) satisfies with probability $1-o(1)$ that every
red-blue-coloring of its edges contains a red copy of $G$ or a blue copy of
$H$. We prove a general upper bound on the threshold for this property under
the assumption that the denser of the two graphs satisfies a certain
balancedness condition. Our result partially confirms a conjecture by the first
author and Kreuter, and together with earlier lower bound results establishes
the exact order of magnitude of the threshold for the case in which $G$ and $H$
are complete graphs of arbitrary size.
  In our proof we present an alternative to the so-called deletion method,
which was introduced by R\"odl and Ruci\'{n}ski in their study of symmetric
Ramsey properties of random graphs (i.e. the case $G=H$), and has been used in
many proofs of similar results since then
Random walks which prefer unvisited edges : exploring high girth even degree expanders in linear time.

Let G = (V,E) be a connected graph with |V | = n vertices. A simple random walk on the vertex set of G is a process, which at each step moves from its current vertex position to a neighbouring vertex chosen uniformly at random. We consider a modified walk which, whenever possible, chooses an unvisited edge for the next transition; and makes a simple random walk otherwise. We call such a walk an edge-process (or E -process). The rule used to choose among unvisited edges at any step has no effect on our analysis. One possible method is to choose an unvisited edge uniformly at random, but we impose no such restriction. For the class of connected even degree graphs of constant maximum degree, we bound the vertex cover time of the E -process in terms of the edge expansion rate of the graph G, as measured by eigenvalue gap 1 -λmax of the transition matrix of a simple random walk on G. A vertex v is ℓ -good, if any even degree subgraph containing all edges incident with v contains at least ℓ vertices. A graph G is ℓ -good, if every vertex has the ℓ -good property. Let G be an even degree ℓ -good expander of bounded maximum degree. Any E -process on G has vertex cover time \ud
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equation image\ud
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This is to be compared with the Ω(nlog n) lower bound on the cover time of any connected graph by a weighted random walk. Our result is independent of the rule used to select the order of the unvisited edges, which could, for example, be chosen on-line by an adversary. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2013\ud
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As no walk based process can cover an n vertex graph in less than n - 1 steps, the cover time of the E -process is of optimal order when ℓ =Θ (log n). With high probability random r -regular graphs, r ≥ 4 even, have ℓ =Ω (log n). Thus the vertex cover time of the E -process on such graphs is Θ(n)
The Janson inequalities for general up-sets

Janson and Janson, Luczak and Rucinski proved several inequalities for the
lower tail of the distribution of the number of events that hold, when all the
events are up-sets (increasing events) of a special form - each event is the
intersection of some subset of a single set of independent events (i.e., a
principal up-set). We show that these inequalities in fact hold for arbitrary
up-sets, by modifying existing proofs to use only positive correlation,
avoiding the need to assume positive correlation conditioned on one of the
events.Comment: 5 pages. Added weighted varian
Strong spatial mixing of list coloring of graphs

The property of spatial mixing and strong spatial mixing in spin systems has been of interest because of its implications on uniqueness of Gibbs measures on infinite graphs and efficient approximation of counting problems that are otherwise known to be #P hard. In the context of coloring, strong spatial mixing has been established for Kelly trees in (Ge and Stefankovic, arXiv:1102.2886v3 (2011)) when q ≥ α[superscript *] Δ + 1 where q the number of colors, Δ is the degree and α[superscript *] is the unique solution to xe[superscript -1/x] = 1. It has also been established in (Goldberg et al., SICOMP 35 (2005) 486–517) for bounded degree lattice graphs whenever q ≥ α[superscript *] Δ - β for some constant β, where Δ is the maximum vertex degree of the graph. We establish strong spatial mixing for a more general problem, namely list coloring, for arbitrary bounded degree triangle-free graphs. Our results hold for any α > α[superscript *] whenever the size of the list of each vertex v is at least αΔ(v) + β where Δ(v) is the degree of vertex v and β is a constant that only depends on α. The result is obtained by proving the decay of correlations of marginal probabilities associated with graph nodes measured using a suitably chosen error function
Efficient algorithms for three-dimensional axial and planar random assignment problems

Beautiful formulas are known for the expected cost of random two-dimensional assignment problems, but in higher dimensions even the scaling is not known. In three dimensions and above, the problem has natural “Axial” and “Planar” versions, both of which are NP-hard. For 3-dimensional Axial random assignment instances of size n, the cost scales as Ω(1/ n), and a main result of the present paper is a linear-time algorithm that, with high probability, finds a solution of cost O(n–1+o(1)). For 3-dimensional Planar assignment, the lower bound is Ω(n), and we give a new efficient matching-based algorithm that with high probability returns a solution with cost O(n log n)
Hypergraph limits: a regularity approach

A sequence of $k$-uniform hypergraphs $H_1, H_2, \dots$ is convergent if the
sequence of homomorphism densities $t(F, H_1), t(F, H_2), \dots$ converges for
every $k$-uniform hypergraph $F$. For graphs, Lov\'asz and Szegedy showed that
every convergent sequence has a limit in the form of a symmetric measurable
function $W \colon [0,1]^2 \to [0,1]$. For hypergraphs, analogous limits $W
\colon [0,1]^{2^k-2} \to [0,1]$ were constructed by Elek and Szegedy using
ultraproducts. These limits had also been studied earlier by Hoover, Aldous,
and Kallenberg in the setting of exchangeable random arrays.
  In this paper, we give a new proof and construction of hypergraph limits. Our
approach is inspired by the original approach of Lov\'asz and Szegedy, with the
key ingredient being a weak Frieze-Kannan type regularity lemma.Comment: 17 pages, 1 figure. Minor typos correcte
Realisability conditions for second order marginals of biphased media

16 pagesInternational audienceThis paper concerns the second order marginals of biphased random media. We give discriminating necessary conditions for a bivariate function to be such a valid marginal, and illustrate our study with two practical applications: (1) the spherical variograms are valid indicator variograms if and only if they are multiplied by a sufficiently small constant, which upper bound is estimated, and (2) not every covariance/indicator variogram can be obtained with a Gaussian level set. The theoretical results backing this study are contained in a companion paper
On independent sets in random graphs

The independence number of a sparse random graph G(n,m) of average degree
d=2m/n is well-known to be \alpha(G(n,m))~2n ln(d)/d with high probability.
Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size
(1+o(1)) n ln(d)/d, i.e. half the maximum size. Yet in spite of 30 years of
extensive research no efficient algorithm has emerged to produce an independent
set with (1+c)n ln(d)/d, for any fixed c>0. In this paper we prove that the
combinatorial structure of the independent set problem in random graphs
undergoes a phase transition as the size k of the independent sets passes the
point k nln(d)/d. Roughly speaking, we prove that independent sets of size
k>(1+c)n ln(d)/d form an intricately ragged landscape, in which local search
algorithms are bound to get stuck. We illustrate this phenomenon by providing
an exponential lower bound for the Metropolis process, a Markov chain for
sampling independents sets.Comment: Polished previous versio
Group representations that resist random sampling

We show that there exists a family of groups $G_n$ and nontrivial irreducible
representations $\rho_n$ such that, for any constant $t$, the average of
$\rho_n$ over $t$ uniformly random elements $g_1, \ldots, g_t \in G_n$ has
operator norm $1$ with probability approaching 1 as $n \rightarrow \infty$.
More quantitatively, we show that there exist families of finite groups for
which $\Omega(\log \log |G|)$ random elements are required to bound the norm of
a typical representation below $1$. This settles a conjecture of A. Wigderson
Ramanujan graphings and correlation decay in local algorithms

Let $G$ be a large-girth $d$-regular graph and $\mu$ be a random process on
the vertices of $G$ produced by a randomized local algorithm. We prove the
upper bound $(k+1-2k/d)\Bigl(\frac{1}{\sqrt{d-1}}\Bigr)^k$ for the (absolute
value of the) correlation of values on pairs of vertices of distance $k$ and
show that this bound is optimal. The same results hold automatically for factor
of i.i.d processes on the $d$-regular tree. In that case we give an explicit
description for the (closure) of all possible correlation sequences. Our proof
is based on the fact that the Bernoulli graphing of the infinite $d$-regular
tree has spectral radius $2\sqrt{d-1}$. Graphings with this spectral gap are
infinite analogues of finite Ramanujan graphs and they are interesting on their
own right
A sprouting tree model for random boolean functions

We define a new probability distribution for Boolean functions of k variables. Consider the random Binary Search Tree of size n, and label its internal nodes by connectives and its leaves by variables or their negations. This random process defines a random Boolean expression which represents a random Boolean function. Finally, let n tend to infinity: the asymptotic distribution on Boolean functions exists; we call it the sprouting tree distribution. We study this model and compare it with two previously-known distributions induced by two other random trees: the Catalan tree and the Galton-Watson tree
Mixing of the Glauber dynamics for the ferromagnetic Potts model.

We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree (Δ) of the underlying graph and the number of colours or spins (q) in determining whether the dynamics mixes rapidly or not. We find a lower bound L on the number of colours such that Glauber dynamics is rapidly mixing if at least L colours are used. We give a closely-matching upper bound U on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random Δ-regular graphs when at most U colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree Δ, e.g. toroidal grids for Δ = 4
The lower tail: Poisson approximation revisited

The well-known "Janson's inequality" gives Poisson-like upper bounds for the
lower tail probability \Pr(X \le (1-\eps)\E X) when X is the sum of dependent
indicator random variables of a special form. We show that, for large
deviations, this inequality is optimal whenever X is approximately Poisson,
i.e., when the dependencies are weak. We also present correlation-based
approaches that, in certain symmetric applications, yield related conclusions
when X is no longer close to Poisson. As an illustration we, e.g., consider
subgraph counts in random graphs, and obtain new lower tail estimates,
extending earlier work (for the special case \eps=1) of Janson, Luczak and
Rucinski.Comment: 21 page
Asymptotic distribution for the birthday problem with multiple
  coincidences, via an embedding of the collision process

We study the random variable B(c,n), which counts the number of balls that
must be thrown into n equally-sized bins in order to obtain c collisions. The
asymptotic expected value of B(1,n) is the well-known $\sqrt{n\pi/2}$ appearing
in the solution to the birthday problem; the limit distribution and asymptotic
moments of B(1,n) are also well known. We calculate the distribution and
moments of B(c,n) asymptotically as n goes to infinity and c = O(n).
  Our main tools are an embedding of the collision process, realizing the
process as a deterministic function of the standard Poisson process, and a
central limit result by Renyi
Spanning embeddings of arrangeable graphs with sublinear bandwidth

The Bandwidth Theorem of Böttcher, et al. [Mathematische Annalen 343 (2009), 175–205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this result to a-arrangeable graphs H with inline image, where n is the number of vertices of H. Our result implies that sufficiently large n-vertex graphs G with minimum degree at least inline image contain almost all planar graphs on n vertices as subgraphs. Using techniques developed by Allen, et al. [Combinatorica 33 (2013), 125–160] we can also apply our methods to show that almost all planar graphs H have Ramsey number at most inline image. We obtain corresponding results for graphs embeddable on different orientable surface
On giant components and treewidth in the layers model

Given an undirected $n$-vertex graph $G(V,E)$ and an integer $k$, let
$T_k(G)$ denote the random vertex induced subgraph of $G$ generated by ordering
$V$ according to a random permutation $\pi$ and including in $T_k(G)$ those
vertices with at most $k-1$ of their neighbors preceding them in this order.
The distribution of subgraphs sampled in this manner is called the \emph{layers
model with parameter} $k$. The layers model has found applications in studying
$\ell$-degenerate subgraphs, the design of algorithms for the maximum
independent set problem, and in bootstrap percolation.
  In the current work we expand the study of structural properties of the
layers model.
  We prove that there are $3$-regular graphs $G$ for which with high
probability $T_3(G)$ has a connected component of size $\Omega(n)$. Moreover,
this connected component has treewidth $\Omega(n)$. This lower bound on the
treewidth extends to many other random graph models. In contrast, $T_2(G)$ is
known to be a forest (hence of treewidth~1), and we establish that if $G$ is of
bounded degree then with high probability the largest connected component in
$T_2(G)$ is of size $O(\log n)$. We also consider the infinite two-dimensional
grid, for which we prove that the first four layers contain a unique infinite
connected component with probability $1$
Improved bounds for the randomized decision tree Complexity of recursive majority

We consider the randomized decision tree complexity of the recursive 3-majority function. We prove a lower bound of inline image for the two-sided-error randomized decision tree complexity of evaluating height h formulae with error inline image. This improves the lower bound of inline image given by Jayram, Kumar, and Sivakumar (STOC'03), and the one of inline image given by Leonardos (ICALP'13). Second, we improve the upper bound by giving a new zero-error randomized decision tree algorithm that has complexity at most inline image. The previous best known algorithm achieved complexity inline image. The new lower bound follows from a better analysis of the base case of the recursion of Jayram et al. The new algorithm uses a novel “interleaving” of two recursive algorithms
Circular law for random matrices with exchangeable entries

An exchangeable random matrix is a random matrix with distribution invariant
under any permutation of the entries. For such random matrices, we show, as the
dimension tends to infinity, that the empirical spectral distribution tends to
the uniform law on the unit disc. This is an instance of the universality
phenomenon known as the circular law, for a model of random matrices with
dependent entries, rows, and columns. It is also a non-Hermitian counterpart of
a result of Chatterjee on the semi-circular law for random Hermitian matrices
with exchangeable entries. The proof relies in particular on a reduction to a
simpler model given by a random shuffle of a rigid deterministic matrix, on
Hermitization, and also on combinatorial concentration of measure and
combinatorial Central Limit Theorem. A crucial step is a polynomial bound on
the smallest singular value of exchangeable random matrices, which may be of
independent interest
Percolation on random recursive trees

We study Bernoulli bond percolation on a random recursive tree of size $n$
with percolation parameter $p(n)$ converging to $1$ as $n$ tends to infinity.
The sizes of the percolation clusters are naturally stored in a tree. We prove
convergence in distribution of this tree to the genealogical tree of a
continuous-state branching process in discrete time. As a corollary we obtain
the asymptotic sizes of the largest and next largest percolation clusters,
extending thereby a recent work of Bertoin (2014) which deals with cluster
sizes in the supercritical regime. In a second part, we show that the same
limit tree appears in the study of the tree components which emerge from a
continuous-time destruction of a random recursive tree. We comment on the
connection to our first result on Bernoulli bond percolation.Comment: 32 pages, 4 figure
Absorption time of the Moran process

© 2016 Wiley Periodicals, Inc.\ud
The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is O(n4) on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we show that the expected absorption time is O(nlogn) and O(n2). We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.Peer ReviewedPostprint (author's final draft
Random points in halfspheres

A random spherical polytope Pn in a spherically convex set K⊂Sd as considered here is the spherical convex hull of n independent, uniformly distributed random points in K. The behaviour of Pn for a spherically convex set K contained in an open halfsphere is quite similar to that of a similarly generated random convex polytope in a Euclidean space, but the case when K is a halfsphere is different. This is what we investigate here, establishing the asymptotic behaviour, as n tends to infinity, of the expectation of several characteristics of Pn, such as facet and vertex number, volume and surface area. For the Hausdorff distance from the halfsphere, we obtain also some almost sure asymptotic estimates. © 2016 Wiley Periodicals, Inc
Constructing near spanning trees with few local inspections

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph math formula consisting of math formula edges (for a prespecified constant math formula), where the decision for different edges should be consistent with the same subgraph math formula. Can this task be performed by inspecting only a constant number of edges in G? Our main results are:
We show that if every t-vertex subgraph of G has expansion math formula then one can (deterministically) construct a sparse spanning subgraph math formula of G using few inspections. To this end we analyze a “local” version of a famous minimum-weight spanning tree algorithm.
We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3-regular graphs of high girth, in which every t-vertex subgraph has expansion math formula. We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth.National Science Foundation (U.S.) (CCF-1217423)National Science Foundation (U.S.) (CCF-1065125)National Science Foundation (U.S.) (CCF-1420692))Israel Science Foundation (1536/14
Exploring hypergraphs with martingales

Recently, we adapted exploration and martingale arguments of Nachmias and
Peres, in turn based on ideas of Martin-L\"of, Karp and Aldous, to prove
asymptotic normality of the number $L_1$ of vertices in the largest component
$C$ of the random $r$-uniform hypergraph throughout the supercritical regime.
In this paper we take these arguments further to prove two new results: strong
tail bounds on the distribution of $L_1$, and joint asymptotic normality of
$L_1$ and the number $M_1$ of edges of $C$. These results are used in a
separate paper "Counting connected hypergraphs via the probabilistic method" to
enumerate sparsely connected hypergraphs asymptotically.Comment: 32 pages; significantly expanded presentation. To appear in Random
  Structures and Algorithm