106 research outputs found
Finding any given 2-factor in sparse pseudorandom graphs efficiently
Given an -vertex pseudorandom graph and an -vertex graph with
maximum degree at most two, we wish to find a copy of in , i.e.\ an
embedding so that
for all . Particular instances of this problem include finding a
triangle-factor and finding a Hamilton cycle in . Here, we provide a
deterministic polynomial time algorithm that finds a given in any suitably
pseudorandom graph . The pseudorandom graphs we consider are
-bijumbled graphs of minimum degree which is a constant proportion
of the average degree, i.e.\ . A -bijumbled graph is
characterised through the discrepancy property: for any two sets of vertices and . Our condition
on bijumbledness is within a log factor from being
tight and provides a positive answer to a recent question of Nenadov.
We combine novel variants of the absorption-reservoir method, a powerful tool
from extremal graph theory and random graphs. Our approach is based on that of
Nenadov (\emph{Bulletin of the London Mathematical Society}, to appear) and on
ours (arXiv:1806.01676), together with additional ideas and simplifications.Comment: 21 page
Clique Factors: Extremal and Probabilistic Perspectives
A K_r-factor in a graph G is a collection of vertex-disjoint copies of K_r covering the vertex set of G. In this thesis, we investigate these fundamental objects in three settings that lie at the intersection of extremal and probabilistic combinatorics.
Firstly, we explore pseudorandom graphs. An n-vertex graph is said to be (p,ÎČ)-bijumbled if for any vertex sets A, B â V (G), we have e( A, B) = p| A||B| ± ÎČâ|A||B|. We prove that for any 3 †r â N and c > 0 there exists an Δ > 0 such that any n-vertex (p, ÎČ)-bijumbled graph with n â rN, ÎŽ(G) â„ c p n and ÎČ â€ Î” p^{r â1} n, contains a K_r -factor. This implies a corresponding result for the stronger pseudorandom notion of (n, d, λ)-graphs. For the case of K_3-factors, this result resolves a conjecture of Krivelevich, Sudakov and SzabĂł from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result, we can conclude that the same condition of ÎČ = o( p^2n) actually guarantees that a (p, ÎČ)-bijumbled graph G contains every graph on n vertices with maximum degree at most 2.
Secondly, we explore the notion of robustness for K_3-factors. For a graph G and p â [0, 1], we denote by G_p the random sparsification of G obtained by keeping each edge of G independently, with probability p. We show that there exists a C > 0 such that if p â„ C (log n)^{1/3}n^{â2/3} and G is an n-vertex graph with n â 3N and ÎŽ(G) â„ 2n/3 , then with high probability G_p contains a K_3-factor. Both the minimum degree condition and the probability condition, up to the choice of C, are tight. Our result can be viewed as a common strengthening of the classical extremal theorem of CorrĂĄdi and Hajnal, corresponding to p = 1 in our result, and the famous probabilistic theorem of Johansson, Kahn and Vu establishing the threshold for the appearance of K_3-factors (and indeed all K_r -factors) in G (n, p), corresponding to G = K_n in our result. It also implies a first lower bound on the number of K_3-factors in graphs with minimum degree at least 2n/3, which gets close to the truth.
Lastly, we consider the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze and Martin, where one starts with a dense graph and then adds random edges to it. Specifically, given any fixed 0 < α < 1 â 1/r we determine how many random edges one must add to an n-vertex graph G with ÎŽ(G) ℠α n to ensure that, with high probability, the resulting graph contains a K_r -factor. As one increases α we demonstrate that the number of random edges
required âjumpsâ at regular intervals, and within these intervals our result is best-possible. This work therefore bridges the gap between the seminal work of Johansson, Kahn and Vu mentioned above, which resolves the purely random case, i.e., α = 0, and that of Hajnal and SzemerĂ©di (and CorrĂĄdi and Hajnal for r = 3) showing that when α â„ 1 â 1/r the initial graph already hosts the
desired K_r -factor.Ein K_r -Faktor in einem Graphen G ist eine Sammlung von Knoten-disjunkten Kopien von K_r , die die Knotenmenge von G ĂŒberdecken. Wir untersuchen diese Objekte in drei Kontexten, die an der Schnittstelle zwischen extremaler und probabilistischer Kombinatorik liegen.
Zuerst untersuchen wir Pseudozufallsgraphen. Ein Graph heiĂt (p,ÎČ)-bijumbled, wenn fĂŒr beliebige Knotenmengen A, B â V (G) gilt e( A, B) = p| A||B| ± ÎČâ|A||B|. Wir beweisen, dass es fĂŒr jedes 3 †r â N und c > 0 ein Δ > 0 gibt, so dass jeder n-Knoten (p, ÎČ)-bijumbled Graph mit n â rN, ÎŽ(G) â„ c p n und ÎČ â€ Î” p^{r â1} n, einen K_r -Faktor enthĂ€lt. Dies impliziert ein entsprechendes Ergebnis fĂŒr den stĂ€rkeren Pseudozufallsbegriff von (n, d, λ)-Graphen. Im Fall von K_3-Faktoren, löst dieses Ergebnis eine Vermutung von Krivelevich, Sudakov und SzabĂł aus
dem Jahr 2004 und ist durch eine pseudozufĂ€llige K_3-freie Konstruktion von Alon bestmöglich. TatsĂ€chlich ist in diesem Fall noch mehr wahr: als Korollar dieses Ergebnisses können wir schlieĂen, dass die gleiche Bedingung von ÎČ = o( p^2n) garantiert, dass ein (p, ÎČ)-bijumbled Graph G jeden Graphen mit maximalem Grad 2 enthĂ€lt.
Zweitens untersuchen wir den Begriff der Robustheit fĂŒr K_3-Faktoren. FĂŒr einen Graphen G und p â [0, 1] bezeichnen wir mit G_p die zufĂ€llige Sparsifizierung von G, die man erhĂ€lt, indem man jede Kante von G unabhĂ€ngig von den anderen Kanten mit einer Wahrscheinlichkeit p behĂ€lt. Wir zeigen, dass, wenn p â„ C (log n)^{1/3}n^{â2/3} und G ein n-Knoten-Graph mit n â 3N und ÎŽ(G) â„ 2n/3 ist, G_pmit hoher Wahrscheinlichkeit (mhW) einen K_3-Faktor enthĂ€lt. Sowohl die Bedingung des minimalen Grades als auch die Wahrscheinlichkeitsbedingung sind bestmöglich. Unser Ergebnis ist eine VerstĂ€rkung des klassischen extremalen Satzes von CorrĂĄdi und Hajnal, entsprechend p = 1 in unserem Ergebnis, und des berĂŒhmten probabilistischen Satzes von Johansson, Kahn und Vu, der den Schwellenwert fĂŒr das Auftreten eines K_3-Faktors (und aller K_r -Faktoren) in G (n, p) festlegt, entsprechend G = K_n in unserem Ergebnis. Es impliziert auch eine erste untere Schranke fĂŒr die Anzahl der K_3-Faktoren in Graphen mit einem minimalen Grad von mindestens 2n/3, die der Wahrheit nahe kommt.
SchlieĂlich betrachten wir die Situation von zufĂ€llig gestörten Graphen; ein Modell, bei dem man mit einem dichten Graphen beginnt und dann zufĂ€llige Kanten hinzufĂŒgt. Wir bestimmen, bei gegebenem 0 < α < 1 â 1/r, wie viele zufĂ€llige Kanten man zu einem n-Knoten-Graphen G mit ÎŽ(G) ℠α n hinzufĂŒgen muss, um sicherzustellen, dass der resultierende Graph mhW einen K_r -Faktor enthĂ€lt. Wir zeigen, dass, wenn man α erhöht, die Anzahl der benötigten Zufallskanten in regelmĂ€Ăigen AbstĂ€nden âspringt", und innerhalb dieser AbstĂ€nde unser Ergebnis bestmöglich ist. Diese Arbeit schlieĂt somit die LĂŒcke zwischen der oben erwĂ€hnten bahnbrechenden Arbeit von Johansson, Kahn und Vu, die den rein zufĂ€lligen Fall, d.h. α = 0, löst, und der Arbeit von Hajnal und SzemerĂ©di (und CorrĂĄdi und Hajnal fĂŒr r = 3), die zeigt, dass der ursprĂŒngliche Graph bereits den gewĂŒnschten K_r -Faktor enthĂ€lt, wenn α â„ 1 â 1/r ist
Subsampled Power Iteration: a Unified Algorithm for Block Models and Planted CSP's
We present an algorithm for recovering planted solutions in two well-known
models, the stochastic block model and planted constraint satisfaction
problems, via a common generalization in terms of random bipartite graphs. Our
algorithm matches up to a constant factor the best-known bounds for the number
of edges (or constraints) needed for perfect recovery and its running time is
linear in the number of edges used. The time complexity is significantly better
than both spectral and SDP-based approaches.
The main contribution of the algorithm is in the case of unequal sizes in the
bipartition (corresponding to odd uniformity in the CSP). Here our algorithm
succeeds at a significantly lower density than the spectral approaches,
surpassing a barrier based on the spectral norm of a random matrix.
Other significant features of the algorithm and analysis include (i) the
critical use of power iteration with subsampling, which might be of independent
interest; its analysis requires keeping track of multiple norms of an evolving
solution (ii) it can be implemented statistically, i.e., with very limited
access to the input distribution (iii) the algorithm is extremely simple to
implement and runs in linear time, and thus is practical even for very large
instances
Complexity Theory
Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical ïŹelds such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes
Pseudorandom hypergraph matchings
A celebrated theorem of Pippenger states that any almost regular hypergraph
with small codegrees has an almost perfect matching. We show that one can find
such an almost perfect matching which is `pseudorandom', meaning that, for
instance, the matching contains as many edges from a given set of edges as
predicted by a heuristic argument.Comment: 14 page
A proof of the Erd\H{o}s-Faber-Lov\'asz conjecture
The Erd\H{o}s-Faber-Lov\'{a}sz conjecture (posed in 1972) states that the
chromatic index of any linear hypergraph on vertices is at most . In
this paper, we prove this conjecture for every large . We also provide
stability versions of this result, which confirm a prediction of Kahn.Comment: 39 pages, 2 figures; this version includes additional references and
makes two small corrections (definition of a useful pair in Section 5 and an
additional condition in the statement of Lemma 6.2
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