73 research outputs found
Another approach to non-repetitive colorings of graphs of bounded degree
We propose a new proof technique that aims to be applied to the same problems
as the Lov\'asz Local Lemma or the entropy-compression method. We present this
approach in the context of non-repetitive colorings and we use it to improve
upper-bounds relating different non-repetitive numbers to the maximal degree of
a graph. It seems that there should be other interesting applications to the
presented approach.
In terms of upper-bound our approach seems to be as strong as
entropy-compression, but the proofs are more elementary and shorter. The
application we provide in this paper are upper bounds for graphs of maximal
degree at most : a minor improvement on the upper-bound of the
non-repetitive number, a upper-bound on the weak total
non-repetitive number and a
upper-bound on the total non-repetitive number of graphs. This last result
implies the same upper-bound for the non-repetitive index of graphs, which
improves the best known bound
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Stochastic Local Search and the Lovasz Local Lemma
Stochastic Local Search and the Lovasz Local LemmabyFotios IliopoulosDoctor of Philosophy in Computer ScienceUniversity of California, BerkeleyProfessor Alistair Sinclair, ChairThis thesis studies randomized local search algorithms for finding solutions of constraint satisfaction problems inspired by and extending the Lovasz Local Lemma (LLL). The LLL is a powerful probabilistic tool for establishing the existence of objects satisfying certain properties (constraints). As a probability statement it asserts that, given a family of âbadâ events, if each bad event is individually not very likely and independent of all but a small number of other bad events, then the probability of avoiding all bad events is strictly positive. In a celebrated breakthrough, Moser and Tardos made the LLL constructive for any product probability measure over explicitly presented variables. Specifically, they proved that whenever the LLL condition holds, their Resample algorithm, which repeatedly selects any occurring bad event and resamples all its variables according to the measure, quickly converges to an object with desired properties. In this dissertation we present a framework that extends the work of Moser and Tardos and can be used to analyze arbitrary, possibly complex, focused local search algorithms, i.e., search algorithms whose process for addressing violated constraints, while local, is more sophisticated than obliviously resampling their variables independently of the current configuration. We give several applications of this framework, notably a new vertex coloring algorithm for graphs with sparse vertex neighborhoods that uses a number of colors that matches the algorithmic barrier for random graphs, and polynomial time algorithms for the celebrated (non-constructive) results of Kahn for the Goldberg-Seymour and List-Edge-Coloring Conjectures.Finally, we introduce a generalization of Kolmogorovâs notion of commutative algorithms, cast as matrix commutativity, and show that their output distribution approximates the so-called âLLL-distributionâ, i.e., the distribution obtained by conditioning on avoiding all bad events. This fact allows us to consider questions such as the number of possible distinct final states and the probability that certain portions of the state space are visited by a local search algorithm, extending existing results for the Moser-Tardos algorithm to commutative algorithms
Coloring problems in combinatorics and descriptive set theory
In this dissertation we study problems related to colorings of combinatorial structures both in the âclassicalâ finite context and in the framework of descriptive set theory, with applications to topological dynamics and ergodic theory. This work consists of two parts, each of which is in turn split into a number of chapters. Although the individual chapters are largely independent from each other (with the exception of Chapters 4 and 6, which partially rely on some of the results obtained in Chapter 3), certain common themes feature throughoutâmost prominently, the use of probabilistic techniques.
In Chapter 1, we establish a generalization of the LovĂĄsz Local Lemma (a powerful tool in probabilistic combinatorics), which we call the Local Cut Lemma, and apply it to a variety of problems in graph coloring.
In Chapter 2, we study DP-coloring (also known as correspondence coloring)âan extension of list
coloring that was recently introduced by DvorĂĄk and Postle. The goal of that chapter is to gain some
understanding of the similarities and the differences between DP-coloring and list coloring, and we find many instances of both.
In Chapter 3, we adapt the LovĂĄsz Local Lemma for the needs of descriptive set theory and use it to
establish new bounds on measurable chromatic numbers of graphs induced by group actions.
In Chapter 4, we study shift actions of countable groups on spaces of the form A, where A is a finite set, and apply the LovĂĄsz Local Lemma to find âlargeâ closed shift-invariant subsets X A on which the induced action of is free.
In Chapter 5, we establish precise connections between certain problems in graph theory and in descriptive set theory. As a corollary of our general result, we obtain new upper bounds on Baire measurable chromatic numbers from known results in finite combinatorics.
Finally, in Chapter 6, we consider the notions of weak containment and weak equivalence of probability measure-preserving actions of a countable groupârelations introduced by Kechris that are combinatorial in spirit and involve the way the action interacts with finite colorings of the underlying probability space.
This work is based on the following papers and preprints: [Ber16a; Ber16b; Ber16c; Ber17a; Ber17b;
Ber17c; Ber18a; Ber18b], [BK16; BK17a] (with Alexandr Kostochka), [BKP17] (with Alexandr Kostochka and Sergei Pron), and [BKZ17; BKZ18] (with Alexandr Kostochka and Xuding Zhu)
Algorithmic Graph Theory
The main focus of this workshop was on mathematical techniques needed for the development of eïŹcient solutions and algorithms for computationally diïŹcult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions
Non-linear Log-Sobolev inequalities for the Potts semigroup and applications to reconstruction problems
Consider a Markov process with state space , which jumps continuously to
a new state chosen uniformly at random and regardless of the previous state.
The collection of transition kernels (indexed by time ) is the Potts
semigroup. Diaconis and Saloff-Coste computed the maximum of the ratio of the
relative entropy and the Dirichlet form obtaining the constant in
the -log-Sobolev inequality (-LSI). In this paper, we obtain the best
possible non-linear inequality relating entropy and the Dirichlet form (i.e.,
-NLSI, ). As an example, we show . The more precise NLSIs have been shown by Polyanskiy and Samorodnitsky to
imply various geometric and Fourier-analytic results.
Beyond the Potts semigroup, we also analyze Potts channels -- Markov
transition matrices constant on and off diagonal. (Potts
semigroup corresponds to a (ferromagnetic) subset of matrices with positive
second eigenvalue). By integrating the -NLSI we obtain the new strong data
processing inequality (SDPI), which in turn allows us to improve results on
reconstruction thresholds for Potts models on trees. A special case is the
problem of reconstructing color of the root of a -colored tree given
knowledge of colors of all the leaves. We show that to have a non-trivial
reconstruction probability the branching number of the tree should be at least
This extends previous
results (of Sly and Bhatnagar et al.) to general trees, and avoids the need for
any specialized arguments. Similarly, we improve the state-of-the-art on
reconstruction threshold for the stochastic block model with balanced
groups, for all . These improvements advocate information-theoretic
methods as a useful complement to the conventional techniques originating from
the statistical physics
Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)
We survey work on coloring, list coloring, and painting squares of graphs; in
particular, we consider strong edge-coloring. We focus primarily on planar
graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography,
comments are welcome, published as a Dynamic Survey in Electronic Journal of
Combinatoric
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