Distributed Deterministic Edge Coloring using Bounded Neighborhood Independence

We study the {edge-coloring} problem in the message-passing model of distributed computing. This is one of the most fundamental and well-studied problems in this area. Currently, the best-known deterministic algorithms for (2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01}, where Delta is the maximum degree of the input graph. Also, recent results of \cite{BE10} for vertex-coloring imply that one can get an O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an arbitrarily small constant epsilon > 0. In this paper we devise a drastically faster deterministic edge-coloring algorithm. Specifically, our algorithm computes an O(Delta)-edge-coloring in O(Delta^{epsilon}) + log-star n time, and an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta) + log-star n time. This result improves the previous state-of-the-art {exponentially} in a wide range of Delta, specifically, for 2^{Omega(\log-star n)} \leq Delta \leq polylog(n). In addition, for small values of Delta our deterministic algorithm outperforms all the existing {randomized} algorithms for this problem. On our way to these results we study the {vertex-coloring} problem on the family of graphs with bounded {neighborhood independence}. This is a large family, which strictly includes line graphs of r-hypergraphs for any r = O(1), and graphs of bounded growth. We devise a very fast deterministic algorithm for vertex-coloring graphs with bounded neighborhood independence. This algorithm directly gives rise to our edge-coloring algorithms, which apply to {general} graphs. Our main technical contribution is a subroutine that computes an O(Delta/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p^2) + \log-star n time, for a parameter p, 1 \leq p \leq Delta

Vertex arboricity of triangle-free graphs

Master's Project (M.S.) University of Alaska Fairbanks, 2016The vertex arboricity of a graph is the minimum number of colors needed to color the vertices so that the subgraph induced by each color class is a forest. In other words, the vertex arboricity of a graph is the fewest number of colors required in order to color a graph such that every cycle has at least two colors. Although not standard, we will refer to vertex arboricity simply as arboricity. In this paper, we discuss properties of chromatic number and k-defective chromatic number and how those properties relate to the arboricity of trianglefree graphs. In particular, we find bounds on the minimum order of a graph having arboricity three. Equivalently, we consider the largest possible vertex arboricity of triangle-free graphs of fixed order

Defective Coloring on Classes of Perfect Graphs

In Defective Coloring we are given a graph $G$ and two integers $\chi_d$, $\Delta^*$ and are asked if we can $\chi_d$-color $G$ so that the maximum degree induced by any color class is at most $\Delta^*$. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters $\chi_d$, $\Delta^*$ is set to the smallest possible fixed value that does not trivialize the problem ($\chi_d = 2$ or $\Delta^* = 1$). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either $\chi_d$ or $\Delta^*$ is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both $\chi_d$ and $\Delta^*$ are unbounded

Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

GROTESQUE: Noisy Group Testing (Quick and Efficient)

Group-testing refers to the problem of identifying (with high probability) a (small) subset of $D$ defectives from a (large) set of $N$ items via a "small" number of "pooled" tests. For ease of presentation in this work we focus on the regime when D = \cO{N^{1-\gap}} for some \gap > 0. The tests may be noiseless or noisy, and the testing procedure may be adaptive (the pool defining a test may depend on the outcome of a previous test), or non-adaptive (each test is performed independent of the outcome of other tests). A rich body of literature demonstrates that $\Theta(D\log(N))$ tests are information-theoretically necessary and sufficient for the group-testing problem, and provides algorithms that achieve this performance. However, it is only recently that reconstruction algorithms with computational complexity that is sub-linear in $N$ have started being investigated (recent work by \cite{GurI:04,IndN:10, NgoP:11} gave some of the first such algorithms). In the scenario with adaptive tests with noisy outcomes, we present the first scheme that is simultaneously order-optimal (up to small constant factors) in both the number of tests and the decoding complexity (\cO{D\log(N)} in both the performance metrics). The total number of stages of our adaptive algorithm is "small" (\cO{\log(D)}). Similarly, in the scenario with non-adaptive tests with noisy outcomes, we present the first scheme that is simultaneously near-optimal in both the number of tests and the decoding complexity (via an algorithm that requires \cO{D\log(D)\log(N)} tests and has a decoding complexity of {${\cal O}(D(\log N+\log^{2}D))$}. Finally, we present an adaptive algorithm that only requires 2 stages, and for which both the number of tests and the decoding complexity scale as {${\cal O}(D(\log N+\log^{2}D))$}. For all three settings the probability of error of our algorithms scales as \cO{1/(poly(D)}.Comment: 26 pages, 5 figure

Предсказательное техническое обслуживание трубопроводов на основе экспресс-оценки степени опасности дефектов

The paper describes a tested and proven practical methodology of predictive maintenance of pipelines with two types of defects — “loss of metal” and “pipe wall lamination”, detected by the ILI technology. The laminations are caused by the steel and pipe manufacturing technology, and may also appear during pipeline operation. The laminations can be further classified as metallurgical laminations, hydrogen induced cracking (HIC), non-metallic inclusions, and such. For the defects of the “pipe wall lamination” type the assessment of their level of danger is conducted only after they are converted to surface “loss of metal” type defects. The paper presents models on how to adequately convert the “pipe wall lamination” type of defects to the “loss of metal” type defects. A methodology is described on how to rank the defects according to their level of danger (with respect to the rupture type of failure), and how to perform the probabilistic assessment of the pipeline residual life. In order to account for “leak” and “rupture” types of failure, a computer based express assessment is developed of the level of severity of each defect. This defect assessment is based on graphs, which restrict the permissible sizes of defects and allow making operative decisions as to which maintenance measures should be taken, regarding pipeline segment as a whole. The pipeline defects are ranked according to their potential danger, which depends on their location on the graphs. The probabilistic assessment of the residual pipeline life is performed taking into account the stochastic nature of defect growth. In order to achieve this, the maximal γ-percentile corrosion rate is defined over all detected defects. As the main decision parameter the gamma-percent operating time is chosen. It is characterized by: the safe operating time, and the percentile probability that during this time the pipeline limit state will not be reached. A detailed example of implementation of the described methodology to a real product pipeline segment operating in a severe corrosion environment is given. The economical effect of the implementation is outlined.Статья описывает проверенную практическую методологию предсказательного технического обслуживания (мейнтенанса) трубопроводов с двумя типами дефектов — «потеря металла» и «несплошность металла стенки трубы — расслоение», обнаруженных с помощью технологии внутритрубной диагностики (ВТД). Несплошности металла стенки трубы возникают в процессе сталеплавильного и прокатного производства, а также в процессе эксплуатации. К этим дефектам относятся: металлургические расслоения, водородные расслоения, закаты и плотные неметаллические включения. Для дефектов, относящихся к этому типу, оценка степени опасности производится только после приведения дефекта к поверхностному дефекту типа «потеря металла». В работе представлены модели приведения несплошности металла к поверхностным дефектам типа «потеря металла». Методика описывает способ ранжировки дефектов по уровню их опасности (относительно отказа типа «разрыв») и вероятностную оценку остаточного ресурса трубопровода. Для учета обоих сценариев отказа «течь» и «разрыв» строится компьютерная экспресс-оценка степени опасности дефектных участков трубопровода путем построения графиков, ограничивающих размеры дефектов трубопровода и позволяющих принимать оперативные решения о мерах по дальнейшей эксплуатации трубопровода. Осуществляется классификация потенциальной опасности дефектов трубопровода в зависимости от области их расположения на графиках. Расчет вероятностной оценки прогнозирования остаточного ресурса трубопровода проведен с учетом вероятностного подрастания дефектов. Для этого определяется максимальная, с заданной вероятностью γ, скорость коррозии по всем дефектам. В качестве основного показателя определяется гамма-процентный ресурс, задаваемый двумя численными значениями: наработкой и выраженной в процентах вероятностью того, что в течение этой наработки предельное состояние не будет достигнуто. Данная работа описывает пример применения описанной методологии к наземному участку трубопровода, транспортирующему сильнодействующий коррозионный конденсат. Также обсуждается экономический эффект от реализации представленной методологии

On Derandomizing Local Distributed Algorithms

The gap between the known randomized and deterministic local distributed algorithms underlies arguably the most fundamental and central open question in distributed graph algorithms. In this paper, we develop a generic and clean recipe for derandomizing LOCAL algorithms. We also exhibit how this simple recipe leads to significant improvements on a number of problem. Two main results are: - An improved distributed hypergraph maximal matching algorithm, improving on Fischer, Ghaffari, and Kuhn [FOCS'17], and giving improved algorithms for edge-coloring, maximum matching approximation, and low out-degree edge orientation. The first gives an improved algorithm for Open Problem 11.4 of the book of Barenboim and Elkin, and the last gives the first positive resolution of their Open Problem 11.10. - An improved distributed algorithm for the Lov\'{a}sz Local Lemma, which gets closer to a conjecture of Chang and Pettie [FOCS'17], and moreover leads to improved distributed algorithms for problems such as defective coloring and $k$-SAT.Comment: 37 page