124 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
Directed Steiner Tree and the Lasserre Hierarchy
The goal for the Directed Steiner Tree problem is to find a minimum cost tree
in a directed graph G=(V,E) that connects all terminals X to a given root r. It
is well known that modulo a logarithmic factor it suffices to consider acyclic
graphs where the nodes are arranged in L <= log |X| levels. Unfortunately the
natural LP formulation has a |X|^(1/2) integrality gap already for 5 levels. We
show that for every L, the O(L)-round Lasserre Strengthening of this LP has
integrality gap O(L log |X|). This provides a polynomial time
|X|^{epsilon}-approximation and a O(log^3 |X|) approximation in O(n^{log |X|)
time, matching the best known approximation guarantee obtained by a greedy
algorithm of Charikar et al.Comment: 23 pages, 1 figur
Applications of the LovĂĄsz Local Lemma and related methods
V tĂ©to prĂĄci se zabĂœvĂĄme aplikacemi LovĂĄszova lokĂĄlnĂho lemmatu a s nĂm souvisejĂcĂch metod. PopĂĆĄeme postupnĂœ vĂœvoj tÄchto metod a ukĂĄĆŸeme konkrĂ©tnĂ pĆĂklady jejich uĆŸitĂ na pĆĂkladech z oblasti vĂœzkumu nezĂĄvislĂœch transverzĂĄl a hypergrafĆŻ.ObhĂĄjenoIn this thesis we investigate applications of the LovĂĄsz local lemma and its related methods. We are going to describe the gradual development of these methods and show the specific examples of its use in the field of research on independent transversals and hypergraphs
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)
Digraph Colouring and Arc-Connectivity
The dichromatic number of a digraph is the minimum size of
a partition of its vertices into acyclic induced subgraphs. We denote by
the maximum local edge connectivity of a digraph . Neumann-Lara
proved that for every digraph , . In this
paper, we characterize the digraphs for which . This generalizes an analogue result for undirected graph proved by Stiebitz
and Toft as well as the directed version of Brooks' Theorem proved by Mohar.
Along the way, we introduce a generalization of Haj\'os join that gives a new
way to construct families of dicritical digraphs that is of independent
interest.Comment: 34 pages, 11 figure
On Approximability, Convergence, and Limits of CSP Problems
This thesis studies dense constraint satisfaction problems (CSPs), and other related optimization and decision problems that can be phrased as questions regarding parameters or properties of combinatorial objects such as uniform hypergraphs. We concentrate on the information that can be derived from a very small substructure that is selected uniformly at random. In this thesis, we present a unified framework on the limits of CSPs in the sense of the convergence notion of Lovasz-Szegedy that depends only on the remarkable connection between graph sequences and exchangeable arrays established by Diaconis-Janson. In particular, we formulate and prove a representation theorem for compact colored r-uniform directed hypergraphs and apply this to rCSPs. We investigate the sample complexity of testable r-graph parameters, and discuss a generalized version of ground state energies (GSE) and demonstrate that they are efficiently testable. The GSE is a term borrowed from statistical physics that stands for a generalized version of maximal multiway cut problems from complexity theory, and was studied in the dense graph setting by Borgs et al. A notion related to testing CSPs that are defined on graphs, the nondeterministic property testing, was introduced by Lovasz-Vesztergombi, which extends the graph property testing framework of Goldreich-Goldwasser-Ron in the dense graph model. In this thesis, we study the sample complexity of nondeterministically testable graph parameters and properties and improve existing bounds by several orders of magnitude. Further, we prove the equivalence of the notions of nondeterministic and deterministic parameter and property testing for uniform dense hypergraphs of arbitrary rank, and provide the first effective upper bound on the sample complexity in this general case
Simultaneous Feedback Edge Set: A Parameterized Perspective
In a recent article Agrawal et al. (STACS 2016) studied a simultaneous variant of the classic Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set (Sim-FVS). In this problem the input is an n-vertex graph G, an integer k and a coloring function col : E(G) -> 2^[alpha]and the objective is to check whether there exists a vertex subset S of cardinality at most k in G such that for all i in [alpha], G_i - S is acyclic. Here, G_i = (V (G), {e in E(G) | i in col(e)}) and [alpha] = {1,...,alpha}. In this paper we consider the edge variant of the problem, namely, Simultaneous Feedback Edge Set (Sim-FES). In this problem, the input is same as the input of Sim-FVS and the objective is to check whether there is an edge subset S of cardinality at most k in G such that for all i in [alpha], G_i - S is acyclic. Unlike the vertex variant of the problem, when alpha = 1, the problem is equivalent to finding a maximal spanning forest and hence it is polynomial time solvable. We show that for alpha = 3 Sim-FES is NP-hard by giving a reduction from Vertex Cover on cubic-graphs. The same reduction shows that the problem does not admit an algorithm of running time O(2^o(k) n^O(1)) unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time O(2^((omega k alpha) + (alpha log k)) n^O(1)), where omega is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when alpha = 2. We also give a kernel for Sim-FES with (k alpha)^O(alpha) vertices. Finally, we consider the problem Maximum Simultaneous Acyclic Subgraph. Here, the input is a graph G, an integer q and, a coloring function col : E(G) -> 2^[alpha] . The question is whether there is a edge subset F of cardinality at least q in G such that for all i in [alpha], G[F_i] is acyclic. Here, F_i = {e in F | i in col(e)}. We give an FPT algorithm for Maximum Simultaneous Acyclic Subgraph running in time O(2^(omega q alpha) n^O(1) ). All our algorithms are based on parameterized version of the Matroid Parity problem
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