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We present a simple deterministic distributed algorithm that computes a $(\Delta+1)$-vertex coloring in $O(\log^2 \Delta \cdot \log n)$ rounds. The algorithm can be implemented with $O(\log n)$-bit messages. The algorithm can also be…

Data Structures and Algorithms · Computer Science 2021-09-07 Mohsen Ghaffari , Fabian Kuhn

Brooks' theorem states that all connected graphs but odd cycles and cliques can be colored with $\Delta$ colors, where $\Delta$ is the maximum degree of the graph. Such colorings have been shown to admit non-trivial distributed algorithms…

Distributed, Parallel, and Cluster Computing · Computer Science 2025-03-07 Yann Bourreau , Sebastian Brandt , Alexandre Nolin

This work concerns the analysis and design of distributed first-order optimization algorithms over time-varying graphs. The goal of such algorithms is to optimize a global function that is the average of local functions using only local…

Optimization and Control · Mathematics 2020-02-17 Akhil Sundararajan , Bryan Van Scoy , Laurent Lessard

Linial's famous color reduction algorithm reduces a given $m$-coloring of a graph with maximum degree $\Delta$ to a $O(\Delta^2\log m)$-coloring, in a single round in the LOCAL model. We show a similar result when nodes are restricted to…

Data Structures and Algorithms · Computer Science 2020-07-31 Yannic Maus , Tigran Tonoyan

$ \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

It is a well known fact that sequential algorithms which exhibit a strong "local" nature can be adapted to the distributed setting given a legal graph coloring. The running time of the distributed algorithm will then be at least the number…

Distributed, Parallel, and Cluster Computing · Computer Science 2018-05-15 Ken-ichi Kawarabayashi , Gregory Schwartzman

The distributed coloring problem is arguably one of the key problems studied in the area of distributed graph algorithms. The most standard variant of the problem asks for a proper vertex coloring of a graph with $\Delta+1$ colors, where…

Distributed, Parallel, and Cluster Computing · Computer Science 2026-05-05 Marc Fuchs , Fabian Kuhn

Consider the following simple coloring algorithm for a graph on $n$ vertices. Each vertex chooses a color from $\{1, \dotsc, \Delta(G) + 1\}$ uniformly at random. While there exists a conflicted vertex choose one such vertex uniformly at…

Data Structures and Algorithms · Computer Science 2021-05-04 Daniel Bertschinger , Johannes Lengler , Anders Martinsson , Robert Meier , Angelika Steger , Miloš Trujić , Emo Welzl

We propose a general method for converting online algorithms to local computation algorithms by selecting a random permutation of the input, and simulating running the online algorithm. We bound the number of steps of the algorithm using a…

Data Structures and Algorithms · Computer Science 2012-05-08 Yishay Mansour , Aviad Rubinstein , Shai Vardi , Ning Xie

We study a family of closely-related distributed graph problems, which we call degree splitting, where roughly speaking the objective is to partition (or orient) the edges such that each node's degree is split almost uniformly. Our findings…

Data Structures and Algorithms · Computer Science 2016-08-11 Mohsen Ghaffari , Hsin-Hao Su

We provide new deterministic algorithms for the edge coloring problem, which is one of the classic and highly studied distributed local symmetry breaking problems. As our main result, we show that a $(2\Delta-1)$-edge coloring can be…

Distributed, Parallel, and Cluster Computing · Computer Science 2022-06-03 Alkida Balliu , Sebastian Brandt , Fabian Kuhn , Dennis Olivetti

Graph coloring is one of the central problems in distributed graph algorithms. Much of the research on this topic has focused on coloring with $\Delta+1$ colors, where $\Delta$ denotes the maximum degree. Using $\Delta+1$ colors may be…

Data Structures and Algorithms · Computer Science 2017-08-24 Mohsen Ghaffari , Christiana Lymouri

There is a huge difference in techniques and runtimes of distributed algorithms for problems that can be solved by a sequential greedy algorithm and those that cannot. A prime example of this contrast appears in the edge coloring problem:…

Data Structures and Algorithms · Computer Science 2025-05-27 Manuel Jakob , Yannic Maus , Florian Schager

In this paper we study fractional coloring from the angle of distributed computing. Fractional coloring is the linear relaxation of the classical notion of coloring, and has many applications, in particular in scheduling. It was proved by…

Distributed, Parallel, and Cluster Computing · Computer Science 2021-03-15 Nicolas Bousquet , Louis Esperet , François Pirot

Coloring unit-disk graphs efficiently is an important problem in the global and distributed setting, with applications in radio channel assignment problems when the communication relies on omni-directional antennas of the same power. In…

Distributed, Parallel, and Cluster Computing · Computer Science 2023-01-23 Louis Esperet , Sébastien Julliot , Arnaud de Mesmay

The $\Delta$-vertex coloring problem has become one of the prototypical problems for understanding the complexity of local distributed graph problems on constant-degree graphs. The major open problem is whether the problem can be solved…

Distributed, Parallel, and Cluster Computing · Computer Science 2025-04-08 Manuel Jakob , Yannic Maus

Locally finding a solution to symmetry-breaking tasks such as vertex-coloring, edge-coloring, maximal matching, maximal independent set, etc., is a long-standing challenge in distributed network computing. More recently, it has also become…

Data Structures and Algorithms · Computer Science 2017-02-03 Pierre Fraigniaud , Marc Heinrich , Adrian Kosowski

In this paper, we consider algorithms for edge-coloring multigraphs $G$ of bounded maximum degree, i.e., $\Delta(G) = O(1)$. Shannon's theorem states that any multigraph of maximum degree $\Delta$ can be properly edge-colored with…

Data Structures and Algorithms · Computer Science 2023-10-31 Abhishek Dhawan

We obtain better algorithms for computing more balanced orientations and degree splits in LOCAL. Important to our result is a connection to the hypergraph sinkless orientation problem [BMNSU, SODA'25] We design an algorithm of complexity…

Data Structures and Algorithms · Computer Science 2026-04-03 Yannic Maus , Alexandre Nolin , Florian Schager

In this paper, we present improved algorithms for the $(\Delta+1)$ (vertex) coloring problem in the Congested-Clique model of distributed computing. In this model, the input is a graph on $n$ nodes, initially each node knows only its…

Data Structures and Algorithms · Computer Science 2020-01-14 Merav Parter