arXiv subjects

Marc Fuchs

Publications and source records attributed to Marc Fuchs.

6 recordsLinked to original sources

arXiv2026

Greedy-Like Defective Coloring: Distributed Algorithms and Applications

A $d$-defective $c$-coloring of a graph $G=(V,E)$ is a coloring of the nodes $V$ with $c$ colors such that every node has at most $d$ neighbors of the same color. Distributed algorithms for computing different variants of defective coloring are at the core of most deterministic state-of-the-art distributed coloring algorithms, and they are also an important tool in many other distributed graph algorithms. In several cases, the overall complexity could be improved if some version of defective coloring could be solved more efficiently. Barenboim and Elkin [STOC '09] introduced a two-pass greedy algorithm that uses $p^2$ colors with defect $\lfloor \Delta/p\rfloor$ in $O(\Delta+\log^{\ast} n)$ rounds. This remains the best defect/color tradeoff for $O(\log^{\ast} n)$-time algorithms in bounded-degree graphs. This paper expands the capabilities of this two-pass algorithm. First, we generalize it to the \emph{list defective coloring} problem (Fuchs and Kuhn, [DISC '23]). Consequently, we obtain an alternative algorithm for computing a proper $(\Delta+1)$-coloring in $\tilde{O}(\sqrt{\Delta}) + O(\log^{\ast} n)$ rounds in the CONGEST model. Second, we analyze a generalized two-pass algorithm for standard defective colorings. We prove that if the number of colors $c$ is not a perfect square, we can improve the state-of-the-art defect for distributed $c$-colorings by a constant factor in most cases. However, we also prove a limitation: for any $c\geq 1$, this generalized algorithm cannot achieve a $c$-coloring with defect below $(1-o(1))\cdot\Delta/\sqrt{c}$.

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arXiv2025

Distributed $(\Delta+1)$-Coloring in Graphs of Bounded Neighborhood Independence

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 $\Delta$ is the maximum degree of the graph. Despite an immense amount of work on distributed coloring problems in the distributed setting, determining the deterministic complexity of $(\Delta + 1)$-coloring in the standard message passing model remains one of the most important open questions of the area. In this paper, we aim to improve our understanding of the deterministic complexity of $(\Delta + 1)$-coloring as a function of $\Delta$ in a special family of graphs for which significantly faster algorithms are already known. The neighborhood independence $\theta$ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. In general, in graphs of neighborhood independence $\theta = O(1)$ (e.g., line graphs), it is known that $(\Delta + 1)$-coloring can be solved in $2^{O(\sqrt{\log \Delta})} + O(\log^* n)$ rounds. In the present paper, we significantly improve this result, and we show that in graphs of bounded neighborhood independence, a $(\Delta + 1)$-coloring can be computed in $(\log \Delta)^{O(\log \log \Delta / \log \log \log \Delta)} + O(\log^* n)$ rounds and thus in quasipolylogarithmic time in $\Delta$. Additionally, we show that when $\theta = o(\Delta^{1/8})$, one can color the graph in $o(\sqrt{\Delta}) + O(\log^* n)$ rounds, which is faster than the current best known coloring algorithms do on general graphs. We also show that the known approach that leads to a polylogarithmic in $\Delta$ algorithm for $(2\Delta - 1)$-edge coloring already fails for edge colorings of hypergraphs of rank at least 3.

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arXiv2025

Round and Resilience-Optimal Approximate Agreement on Trees and Block Graphs

Approximate Agreement ($\mathcal{AA}$) is a fundamental primitive that, even in the presence of Byzantine faults, allows honest parties to obtain close (but not necessarily identical) outputs that lie within the range of their inputs. While the optimal round complexity of synchronous $\mathcal{AA}$ on real values is well understood, its extension to other input spaces has remained open, with fundamental questions regarding achievable resilience and round efficiency still unresolved. In this work, we investigate the optimal round complexity of synchronous $\mathcal{AA}$ on trees under Byzantine failures. In this setting, parties hold as inputs vertices of a publicly known labeled tree $T$ and must output $1$-close vertices lying in the convex hull of the honest inputs. We present a synchronous protocol with optimal resilience and round complexity $O\left(\frac{\log D(T)}{\log \log D(T)}\right)$, where $D(T)$ denotes the diameter of the input space tree. Complementing this result, we extend impossibility results for real-valued $\mathcal{AA}$ to any graph $G$ by proving a lower bound of $\Omega\left(\frac{\log D(G)}{\log \log D(G) + \log \frac{n+t}{t}}\right)$ rounds, where $n$ is the number of parties and $t$ the number of Byzantine faults. Together, these results establish the asymptotic optimality of our protocol whenever $t \in \Theta(n)$. We further extend our techniques to block graphs by leveraging their clique tree structure. This yields protocols for $\mathcal{AA}$ on block graphs with optimal resilience in both the synchronous and asynchronous models, and with optimal round complexity in the synchronous model.

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arXiv2024

Simpler and More General Distributed Coloring Based on Simple List Defective Coloring Algorithms

In this paper, we give list coloring variants of simple iterative defective coloring algorithms. Formally, in a list defective coloring instance, each node $v$ of a graph is given a list $L_v$ of colors and a list of allowed defects $d_v(x)$ for the colors. Each node $v$ needs to be colored with a color $x\in L_v$ such that at most $d_v(x)$ neighbors of $v$ also pick the same color $x$. For a defect parameter $d$, it is known that by making two sweeps in opposite order over the nodes of an edge-oriented graph with maximum outdegree $\beta$, one can compute a coloring with $O(\beta^2/d^2)$ colors such that every node has at most $d$ outneighbors of the same color. We generalize this and show that if all nodes have lists of size $p^2$ and $\forall v:\sum_{x\in L_v}(d_v(x)+1)>p\cdot\beta$, we can make two sweeps of the nodes such that at the end, each node $v$ has chosen a color $x\in L_v$ for which at most $d_v(x)$ outneighbors of $v$ are colored with color $x$. Our algorithm is simpler and computationally significantly more efficient than existing algorithms for similar list defective coloring problems. We show that the above result can in particular be used to obtain an alternative $\tilde{O}(\sqrt{\Delta})+O(\log^* n)$-round algorithm for the $(\Delta+1)$-coloring problem in the CONGEST model. The neighborhood independence $\theta$ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. It is known that by doing a single sweep over the nodes of a graph of neighborhood independence $\theta$, one can compute a $d$-defective coloring with $O(\theta\cdot \Delta/d)$ colors. We extend this approach to the list defective coloring setting and use it to obtain an efficient recursive coloring algorithm for graphs of neighborhood independence $\theta$. In particular, if $\theta=O(1)$, we get an $(\log\Delta)^{O(\log\log\Delta)}+O(\log^* n)$-round algorithm.

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arXiv2023

List Defective Colorings: Distributed Algorithms and Applications

The distributed coloring problem is at the core of the area of distributed graph algorithms and it is a problem that has seen tremendous progress over the last few years. Much of the remarkable recent progress on deterministic distributed coloring algorithms is based on two main tools: a) defective colorings in which every node of a given color can have a limited number of neighbors of the same color and b) list coloring, a natural generalization of the standard coloring problem that naturally appears when colorings are computed in different stages and one has to extend a previously computed partial coloring to a full coloring. In this paper, we introduce 'list defective colorings', which can be seen as a generalization of these two coloring variants. Essentially, in a list defective coloring instance, each node $v$ is given a list of colors $x_{v,1},\dots,x_{v,p}$ together with a list of defects $d_{v,1},\dots,d_{v,p}$ such that if $v$ is colored with color $x_{v, i}$, it is allowed to have at most $d_{v, i}$ neighbors with color $x_{v, i}$. We highlight the important role of list defective colorings by showing that faster list defective coloring algorithms would directly lead to faster deterministic $(\Delta+1)$-coloring algorithms in the LOCAL model. Further, we extend a recent distributed list coloring algorithm by Maus and Tonoyan [DISC '20]. Slightly simplified, we show that if for each node $v$ it holds that $\sum_{i=1}^p \big(d_{v,i}+1)^2 > \mathrm{deg}_G^2(v)\cdot polylog\Delta$ then this list defective coloring instance can be solved in a communication-efficient way in only $O(\log\Delta)$ communication rounds. This leads to the first deterministic $(\Delta+1)$-coloring algorithm in the standard CONGEST model with a time complexity of $O(\sqrt{\Delta}\cdot polylog \Delta+\log^* n)$, matching the best time complexity in the LOCAL model up to a $polylog\Delta$ factor.

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arXiv2021

Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

We provide CONGEST model algorithms for approximating minimum weighted vertex cover and the maximum weighted matching. For bipartite graphs, we show that a $(1+\varepsilon)$-approximate weighted vertex cover can be computed deterministically in polylogarithmic time. This generalizes a corresponding result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS '20]. Moreover, we show that in general weighted graph families that are closed under taking subgraphs and in which we can compute an independent set of weight at least a $\lambda$-fraction of the total weight, one can compute a $(2-2\lambda +\varepsilon)$-approximate weighted vertex cover in polylogarithmic time in the CONGEST model. Our result in particular implies that in graphs of arboricity $a$, one can compute a $(2-1/a+\varepsilon)$-approximate weighted vertex cover. For maximum weighted matchings, we show that a $(1-\varepsilon)$-approximate solution can be computed deterministically in polylogarithmic CONGEST rounds (for constant $\varepsilon$). We also provide a more efficient randomized algorithm. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie; SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17] for the unweighted case. Finally, we show that even in the LOCAL model and in bipartite graphs of degree $\leq 3$, if $\varepsilon<\varepsilon_0$ for some constant $\varepsilon_0>0$, then computing a $(1+\varepsilon)$-approximation for the unweighted minimum vertex cover problem requires $\Omega\big(\frac{\log n}{\varepsilon}\big)$ rounds. This generalizes aresult of [G\"o\"os, Suomela; DISC '12], who showed that computing a $(1+\varepsilon_0)$-approximation in such graphs requires $\Omega(\log n)$ rounds.

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