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We present a deterministic distributed algorithm, in the LOCAL model, that computes a $(1+o(1))\Delta$-edge-coloring in polylogarithmic-time, so long as the maximum degree $\Delta=\tilde{\Omega}(\log n)$. For smaller $\Delta$, we give a…

Data Structures and Algorithms · Computer Science 2017-11-16 Mohsen Ghaffari , Fabian Kuhn , Yannic Maus , Jara Uitto

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…

Combinatorics · Mathematics 2018-03-22 David R. Wood

Graph coloring involves assigning colors to the vertices of a graph such that two vertices linked by an edge receive different colors. Graph coloring problems are general models that are very useful to formulate many relevant applications…

Machine Learning · Computer Science 2020-10-27 Olivier Goudet , Béatrice Duval , Jin-Kao Hao

We prove new bounds on the distributed fractional coloring problem in the LOCAL model. Fractional $c$-colorings can be understood as multicolorings as follows. For some natural numbers $p$ and $q$ such that $p/q\leq c$, each node $v$ is…

Distributed, Parallel, and Cluster Computing · Computer Science 2021-12-10 Alkida Balliu , Fabian Kuhn , Dennis Olivetti

Combinatorial optimization is a fundamental problem found in many fields. In many real life situations, the constraints and the objective function forming the optimization problem are naturally distributed amongst different sites in some…

Cryptography and Security · Computer Science 2018-03-16 Yuan Hong , Jaideep Vaidya , Haibing Lu

In Defective Coloring we are given a graph $G = (V, E)$ and two integers $\chi_d, \Delta^*$ and are asked if we can partition $V$ into $\chi_d$ color classes, so that each class induces a graph of maximum degree $\Delta^*$. We investigate…

Data Structures and Algorithms · Computer Science 2018-01-12 Rémy Belmonte , Michael Lampis , Valia Mitsou

We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for $(\Delta+1)$- vertex coloring and…

Data Structures and Algorithms · Computer Science 2017-11-15 Sayan Bhattacharya , Deeparnab Chakrabarty , Monika Henzinger , Danupon Nanongkai

We study the problem of bi-chromatic coloring of hypergraphs in the LOCAL distributed model of computation. This problem can easily be solved by a randomized local algorithm with no communication. However, it is not known how to solve it…

Data Structures and Algorithms · Computer Science 2019-08-01 Dariusz R. Kowalski , Piotr Krysta

$ \renewcommand{\tilde}{\widetilde} $We present an $\tilde{O}(\log^2 n)$ round deterministic distributed algorithm for the maximal independent set problem. By known reductions, this round complexity extends also to maximal matching,…

Data Structures and Algorithms · Computer Science 2023-03-29 Mohsen Ghaffari , Christoph Grunau

We consider two models of computation: centralized local algorithms and local distributed algorithms. Algorithms in one model are adapted to the other model to obtain improved algorithms. Distributed vertex coloring is employed to design…

Data Structures and Algorithms · Computer Science 2014-11-12 Guy Even , Moti Medina , Dana Ron

One of the fundamental and most-studied algorithmic problems in distributed computing on networks is graph coloring, both in bounded-degree and in general graphs. Recently, the study of this problem has been extended in two directions.…

Distributed, Parallel, and Cluster Computing · Computer Science 2021-09-14 Nicolas Bousquet , Laurent Feuilloley , Marc Heinrich , Mikaël Rabie

We study the edge-coloring problem in simple $n$-vertex $m$-edge graphs with maximum degree $\Delta$. This is one of the most classical and fundamental graph-algorithmic problems. Vizing's celebrated theorem provides…

Data Structures and Algorithms · Computer Science 2024-07-10 Michael Elkin , Ariel Khuzman

The problem of sampling edge-colorings of graphs with maximum degree $\Delta$ has received considerable attention and efficient algorithms are available when the number of colors is large enough with respect to $\Delta$. Vizing's theorem…

Data Structures and Algorithms · Computer Science 2025-01-22 Lucas De Meyer , František Kardoš , Aurélie Lagoutte , Guillem Perarnau

Graph coloring is a challenging combinatorial optimization problem with a wide range of applications. In this paper, a distribution evolutionary algorithm based on a population of probability model (DEA-PPM) is developed to address it…

Neural and Evolutionary Computing · Computer Science 2023-05-03 Yongjian Xu , Huabin Cheng , Ning Xu , Yu Chen , Chengwang Xie

Irregular computations on unstructured data are an important class of problems for parallel programming. Graph coloring is often an important preprocessing step, e.g. as a way to perform dependency analysis for safe parallel execution. The…

Distributed, Parallel, and Cluster Computing · Computer Science 2015-05-19 Georgios Rokos , Gerard Gorman , Paul H J Kelly

We consider graph coloring and related problems in the distributed message-passing model. {Locally-iterative algorithms} are especially important in this setting. These are algorithms in which each vertex decides about its next color only…

Distributed, Parallel, and Cluster Computing · Computer Science 2021-08-20 Leonid Barenboim , Michael Elkin , Uri Goldenberg

We solve, in a fully decentralised way (\ie with no message passing), the classic problem of colouring a graph. We propose a novel algorithm that is automatically responsive to topology changes, and we prove that it converges quickly to a…

Data Structures and Algorithms · Computer Science 2017-09-05 Alessandro Checco , Douglas J. Leith

We define the $d$-defective incidence chromatic number of a graph, generalizing the notion of incidence chromatic number, and determine it for some classes of graphs including trees, complete bipartite graphs, complete graphs, and…

Combinatorics · Mathematics 2022-02-09 Huimin Bi , Xin Zhang

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvo\v{r}\'{a}k and Postle. Many known upper bounds for the list-chromatic number extend to the DP-chromatic number, but not all…

Combinatorics · Mathematics 2017-11-02 Anton Bernshteyn , Alexandr Kostochka

Given two colorings of a graph, we consider the following problem: can we recolor the graph from one coloring to the other through a series of elementary changes, such that the graph is properly colored after each step? We introduce the…

Distributed, Parallel, and Cluster Computing · Computer Science 2019-01-18 Marthe Bonamy , Paul Ouvrard , Mikaël Rabie , Jukka Suomela , Jara Uitto