arXiv subjects

Yannic Maus

Publications and source records attributed to Yannic Maus.

At least 19 recordsLinked to original sources

arXiv2026

Distributed Symmetry Breaking on Hyperbolic Random Graphs

Real-world networks like the internet share patterns like a power law degree distribution and a high clustering coefficient. Many of these properties are captured by the generative model of hyperbolic random graphs (HRGs), which provides a theoretical framework for studying such networks. Motivated by the observation that several algorithms perform better on real-world networks than their worst-case guarantees suggest, we design and analyse distributed algorithms under the assumption that the input graph is an HRG. Indeed, prior work has shown that the classical symmetry-breaking problem of $\Delta+1$ colouring, where $\Delta$ is the maximum degree of the graph, can be solved in 2 rounds on HRGs [Maus and Ruff; SODA'26]. In stark contrast to this 2-round algorithm for $\Delta+1$ colouring, we prove that the related symmetry-breaking problems of maximal independent set (MIS) and maximal matching (MM) are substantially harder: we establish a lower bound of $\Omega\left(\frac{\log\log n}{\log\log\log n}\right)$ for MIS and MM on HRGs. Our lower bound techniques rely on new structural insights that may be of independent interest: we show that HRGs contain $d$-ary trees with large height and degree which enables us to adapt and lift prior impossibility results for distributed algorithms to the setting of HRGs. We also show that these lower bounds are polynomial tight: we design algorithms tailored to HRGs that solve MIS and MM in $\tilde{\mathcal{O}}(\log^{5/3}\log n)$ rounds with high probability in the LOCAL model, improving over the general worst-case lower bound of $\Omega\left(\min\left\{\log \Delta, \sqrt{\log n}\right\}\right)$ rounds [Khoury and Schild; FOCS'25].

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arXiv2026

Robust Shattering Arguments

Graph shattering is a central technique underlying sublogarithmic-time distributed algorithms in the LOCAL model. Its analysis typically relies on bounding the probability that large sets of distant nodes remain unresolved, often via independence assumptions justified by locality. We show that these assumptions fail for pre-shattering procedures that run for super-constant rounds, where dependencies accumulate over time. As a result, several standard shattering arguments in the literature are incomplete, including those for maximal independent set, $(\Delta+1)$-coloring, and the distributed Lov\'asz Local Lemma (LLL). We provide a systematic repair of these analyses. Our main contribution is a corrected shattering analysis of the Fischer--Ghaffari LLL algorithm. In addition, we develop general tools that capture common patterns in modern algorithms and yield the required decay bounds without relying on independence. We also present explicit counterexamples to commonly used shattering lemmas. Overall, we establish a robust and reusable foundation for shattering arguments in the presence of long-range dependencies.

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arXiv2026

Near-Optimal Distributed 2-Ruling Sets on Graphs with Low Arboricity

Given a graph $G=(V,E)$, a $\beta$-ruling set is a subset of nodes $S\subseteq V$ that is independent, and each node in $V$ is at distance at most $\beta$ from some node in $S$. In this paper, we present almost optimal distributed algorithms for finding $2$-ruling sets in the classical \LOCAL model. Our main contribution is a randomized algorithm that w.h.p.\ computes a $2$-ruling set on any $n$-node graph with bounded arboricity in $O(\log \log n)$ rounds. In fact, the algorithm works up to arboricity $O(\log\log n)$, improves exponentially over the prior state of the art that can be achieved by combining [Barenboim, Elkin, Pettie, Schneider; JACM'16], [Ghaffari; SODA'16], and [Bisht, Kothapalli and Pemmaraju; PODC'14], and nearly matches the lower bound of $\Omega(\log \log n / \log \log \log n)$ [Balliu, Brandt, Kuhn, Olivetti; FOCS'20]. The domination parameter $\beta=2$ is optimal for algorithms with runtime $\log^{o(1)}n$: on graphs with arboricity $2$, there is a lower bound of $\Omega(\sqrt{\log n})$ rounds for MIS (i.e., $\beta = 1$) [Khoury, Schild; FOCS'25]. Additionally, we obtain improved algorithms for larger arboricity. For general graphs with arboricity $\alpha$, we present a randomized algorithm that computes a $2$-ruling set in $\widetilde{O}(\log^{5/8} \alpha +\log^{5/3} \log n)$ rounds. This improves exponentially over the state of the art for a large range of non-constant arboricity. Our techniques extend beyond distributed computing. We present an $O(\log \log \log n)$-round algorithm in the low-space Massively Parallel Computation (\mpc) model that w.h.p.\ computes a $2$-ruling set on any graph with arboricity up to $2^{poly (\log \log n)}$, improving exponentially over the state of the art from [Kothapalli, Pai, Pemmaraju; FSTTCS'20] combined with [Fischer, Giliberti, Grunau; SPAA'23].

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arXiv2026

Deterministic Distance Approximation in MPC via Improved Hitting Sets

In this paper, we provide the first deterministic algorithms with sublogarithmic round complexity for spanners and approximate shortest paths in various MPC models. Moreover, we significantly improve upon the state of the art in the deterministic Congested Clique. In particular, we obtain the following four results on undirected graphs: 1. In both linear MPC and Congested Clique, we obtain an $O(k)$ stretch-spanner of a weighted graph of size $O(n^{1+1/k})$ in $O(1)$ rounds, for some parameter $k\ge 0$. For $k=O(\log{n})$, this leads to an $O(\log n)$ approximation of APSP in constant rounds in both models. 2. In sublinear MPC, we obtain an $O(k^{1+\varepsilon})$-stretch spanner of a weighted graph of size $O(n^{1+1/k})$ in $O(\log k)$ rounds, for any fixed constant $\varepsilon>0$. 3. In Congested Clique, we obtain $O(1)$-approximate APSP for weighted graphs in $O(\log \log \log n)$ rounds. 4. In near-linear MPC, we obtain $(1+\varepsilon)$-approximate single-source shortest paths and $O(1)$-approximate all-pairs shortest paths for unweighted graphs in $\textsf{poly}\log \log n$ rounds. Our algorithm only requires a single near-linear memory machine, where the rest can have sublinear memory. Our deterministic algorithms obtain similar guarantees to the state of the art randomized algorithms without incurring additional factors in the round complexity. To obtain these results, we inspect the randomized algorithms and isolate a randomized sampling routine. Then we derandomize these sampling routines by using a deterministic hitting set. Hereto, we develop a versatile deterministic hitting set algorithm, which we hope will have further derandomization applications.

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arXiv2026

Distributed Santa Claus via Global Rounding

In this paper, we consider the Santa Claus problem in the CONGEST model. This NP-hard problem can be modeled as a bipartite graph of children and gifts where an edge indicates that a child desires a gift. Notably, each gift can have a different value. The goal is to assign the gifts to the children such that the least happy child is as happy as possible. Even though this is a well-studied problem in the sequential setting, we obtain the first results the distributed setting. In particular, we show that the complexity of computing an $\mathcal{O}(\log n/\log \log n)$-approximation is $\hat \Theta(\sqrt n+D)$ rounds, where our $\widetilde\Omega(\sqrt n+D)$-round lower bound is even stronger and holds for any approximation.

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arXiv2026

Near-Optimal Distributed Ruling Sets for Trees and High-Girth Graphs

Given a graph $G=(V,E)$, a $β$-ruling set is a subset $S\subseteq V$ that is i) independent, and ii) every node $v\in V$ has a node of $S$ within distance $β$. In this paper we present almost optimal distributed algorithms for finding ruling sets in trees and high girth graphs in the classic LOCAL model. As our first contribution we present an $O(\log\log n)$-round randomized algorithm for computing $2$-ruling sets on trees, almost matching the $Ω(\log\log n/\log\log\log n)$ lower bound given by Balliu et al. [FOCS'20]. Second, we show that $2$-ruling sets can be solved in $\widetilde{O}(\log^{5/3}\log n)$ rounds in high-girth graphs. Lastly, we show that $O(\log\log\log n)$-ruling sets can be computed in $\widetilde{O}(\log\log n)$ rounds in high-girth graphs matching the lower bound up to triple-log factors. All of these results either improve polynomially or exponentially on the previously best algorithms and use a smaller domination distance $β$.

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arXiv2026

Sublogarithmic Distributed Vertex Coloring with Optimal Number of Colors

For any $Δ$, let $k_Δ$ be the maximum integer $k$ such that $(k+1)(k+2)\le Δ$. We give a distributed \LOCAL algorithm that, given an integer $k < k_Δ$, computes a valid $Δ-k$-coloring if one exists. The algorithm runs in $\tilde{O}(\log^4 \log n)$ rounds, which is within a polynomial factor of the $Ω(\log\log n)$ lower bound, which already applies to the case $k=0$. It is also best possible in the sense that if $k \ge k_Δ$, the problem requires $Ω(n/Δ)$ distributed rounds [Molloy, Reed, '14, Bamas, Esperet '19]. For $Δ$ at most polylogarithmic, the algorithm is an exponential improvement over the current state of the art of $O(\log^{49/12} n)$ rounds. When $Δ\ge (\log n)^{50}$, our algorithm achieves an even faster runtime of $O(\log^* n)$ rounds.

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arXiv2026

Fast Deterministic Distributed Degree Splitting

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 $\mathcal{O}(\varepsilon^{-1} \cdot \log n)$ for computing a balanced orientation with discrepancy at most $\varepsilon \cdot \mathrm{deg}(v)$ for every vertex $v \in V$. This improves upon a previous result by [GHKMSU, Distrib. Comput. 2020] of complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log \varepsilon^{-1} \cdot (\log \log \varepsilon^{-1})^{1.71} \cdot \log n)$. Further, we show that this result can also be extended to compute undirected degree splits with the same discrepancy and in the same runtime. As as application we show that $(3 / 2 + \varepsilon)\Delta$-edge coloring can now be solved in $\mathcal{O}(\varepsilon^{-1} \cdot \log^2 \Delta \cdot \log n + \varepsilon^{-2} \cdot \log n)$ rounds in LOCAL. Note that for constant $\varepsilon$ and $\Delta = \mathcal{O}(2^{\log^{1/3} n})$ this runtime matches the current state-of-the-art for $(2\Delta - 1)$-edge coloring in [Ghaffari & Kuhn, FOCS'21].

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arXiv2026

Deterministic Edge Coloring with few Colors in CONGEST

As the main contribution of this work we present deterministic edge coloring algorithms in the CONGEST model. In particular, we present an algorithm that edge colors any $n$-node graph with maximum degree $\Delta$ with with $(1+\varepsilon)\Delta+O(\sqrt{\log n})$ colors in $\tilde{O}(\log^{2.5} n+\log^2 \Delta \log n)$ rounds. This brings the upper bound polynomially close to the lower bound of $\Omega(\log n/\log\log n)$ rounds that also holds in the more powerful LOCAL model [Chang, He, Li, Pettie, Uitto; SODA'18]. As long as $\Delta \geq c\sqrt{\log n}$ our algorithm uses fewer than $2\Delta-1$ colors and to the best of our knowledge is the first polylogarithmic-round CONGEST algorithm achieving this for any range of $\Delta$. As a corollary we also improve the complexity of edge coloring with $2\Delta-1$ colors for all ranges of $\Delta$ to $\tilde{O}(\log^{2.5} n+\log^2 \Delta \log n)$. This improves upon the previous $O(\log^8 n)$-round algorithm from [Fischer, Ghaffari, Kuhn; FOCS'17]. Our approach builds on a refined analysis and extension of the online edge-coloring algorithm of Blikstad, Svensson, Vintan, and Wajc [FOCS'25], and more broadly on new connections between online and distributed graph algorithms. We show that their algorithm exhibits very low locality and, if it can additionally have limited local access to future edges (as distributed algorithms can), it can be derandomized for smaller degrees. Under this additional power, we are able to bypass classical online lower bounds and translate the results to efficient distributed algorithms. This leads to our CONGEST algorithm for $(1+\varepsilon)\Delta+O(\sqrt{\log n})$-edge coloring. Since the modified online algorithm can be implemented more efficiently in the LOCAL model, we also obtain (marginally) improved complexity bounds in that model.

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arXiv2025

Distributed Sparsest Cut via Eigenvalue Estimation

We give new, improved bounds for approximating the sparsest cut value or in other words the conductance $\phi$ of a graph in the CONGEST model. As our main result, we present an algorithm running in $O(\log^2 n/\phi)$ rounds in which every vertex outputs a value $\tilde \phi$ satisfying $\phi \le \tilde \phi \le \sqrt{2.01\phi}$. In most regimes, our algorithm improves significantly over the previously fastest algorithm for the problem [Chen, Meierhans, Probst Gutenberg, Saranurak; SODA 25]. Additionally, our result generalizes to $k$-way conductance. We obtain these results, by approximating the eigenvalues of the normalized Laplacian matrix $L:=I-{\rm Deg}^{-1/2}A{\rm Deg}^ {-1/2}$, where, $A$ is the adjacency matrix and Deg is the diagonal matrix with the weighted degrees on the diagonal. We show our algorithms are near-optimal by proving a lower bound for computing the smallest non-trivial eigenvalue of $L$, even in the stronger LOCAL model The previous state of the art sparsest cut algorithm is in the technical realm of expander decompositions. Our algorithms, on the other hand, are relatively simple and easy to implement. At the core, they rely on the well-known power method, which comes down to repeatedly multiplying the Laplacian with a vector. This operation can be performed in a single round in the CONGEST model. All our algorithms apply to weighted, undirected graphs. Our lower bounds apply even in unweighted graphs.

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arXiv2025

On Distributed Colouring of Hyperbolic Random Graphs

We analyse the performance of simple distributed colouring algorithms under the assumption that the input graph is a hyperbolic random graph (HRG), a generative model capturing key properties of real-world networks such as power-law degree distributions and large clustering coefficients. Motivated by the shift from worst-case analysis to more realistic network models, we study the number of rounds and size of the colour space required to colour HRGs in the distributed setting.

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arXiv2025

Towards Optimal Distributed Edge Coloring with Fewer Colors

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: while $(2Δ-1)$-edge coloring can be solved in $\mathcal{O}(\log^{\ast}(n))$ rounds on constant-degree graphs, the seemingly minor reduction to $(2Δ-2)$ colors leads to an $Ω(\log n)$ lower bound [Chang, He, Li, Pettie & Uitto, SODA'18]. Understanding this sharp divide between very local problems and inherently more global ones remains a central open question in distributed computing and it is a core focus of this paper. As our main contribution we design a deterministic distributed $\mathcal{O}(\log n)$-round reduction from the $(2Δ-2)$-edge coloring problem to the much easier $(2Δ-1)$-edge coloring problem. This reduction is optimal, as the $(2Δ-2)$-edge coloring problem admits an $Ω(\log n)$ lower bound, whereas the $2Δ-1$-edge coloring problem can be solved in $\mathcal{O}(\log^{\ast}n)$ rounds. By plugging in the $(2Δ-1)$-edge coloring algorithms from [Balliu, Brandt, Kuhn & Olivetti, PODC'22] running in $\mathcal{O}(\log^{12}Δ+ \log^{\ast} n)$ rounds, we obtain an optimal runtime of $\mathcal{O}(\log n)$ rounds as long as $Δ= 2^{\mathcal{O}(\log^{1/12} n)}$. Furthermore, on general graphs our reduction improves the runtime from $\widetilde{\mathcal{O}}(\log^3 n)$ to $\widetilde{\mathcal{O}}(\log^{5/3} n)$. In addition, we also obtain an optimal $\mathcal{O}(\log \log n)$-round randomized reduction of $(2Δ- 2)$-edge coloring to $(2Δ- 1)$-edge coloring. Lastly, we obtain an $\mathcal{O}(\log_Δn)$-round reduction from the $(2Δ-1)$-edge coloring, albeit to the somewhat harder maximal independent set (MIS) problem.

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arXiv2025

Towards Optimal Distributed Delta Coloring

The $Δ$-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 deterministically in logarithmic time, which would match the lower bound [Chang et al., FOCS'16]. Despite recent progress in the design of efficient $Δ$-coloring algorithms, there is currently a polynomial gap between the upper and lower bounds. In this work we present a $O(\log n)$-round deterministic $Δ$-coloring algorithm for dense constant-degree graphs, matching the lower bound for the problem on general graphs. For general $Δ$ the algorithms' complexity is $\min\{\widetilde{O}(\log^{5/3}n),O(Δ+\log n)\}$. All recent distributed and sublinear graph coloring algorithms (also for coloring with more than $Δ$ colors) decompose the graph into sparse and dense parts. Our algorithm works for the case that this decomposition has no sparse vertices. Ironically, in recent (randomized) $Δ$-coloring algorithms, dealing with sparse parts was relatively easy and these dense parts arguably posed the major hurdle. We present a solution that addresses the dense parts and may have the potential for extension to sparse parts. Our approach is fundamentally different from prior deterministic algorithms and hence hopefully contributes towards designing an optimal algorithm for the general case. Additionally, we leverage our result to also obtain a randomized $\min\{\widetilde{O}(\log^{5/3}\log n), O(Δ+\log\log n)\}$-round algorithm for $Δ$-coloring dense graphs that also matches the lower bound for the problem on general constant-degree graphs [Brandt et al.; STOC'16].

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arXiv2025

Hyperbolic Random Graphs: Clique Number and Degeneracy with Implications for Colouring

Hyperbolic random graphs inherit many properties that are present in real-world networks. The hyperbolic geometry imposes a scale-free network with a strong clustering coefficient. Other properties like a giant component, the small world phenomena and others follow. This motivates the design of simple algorithms for hyperbolic random graphs. In this paper we consider threshold hyperbolic random graphs (HRGs). Greedy heuristics are commonly used in practice as they deliver a good approximations to the optimal solution even though their theoretical analysis would suggest otherwise. A typical example for HRGs are degeneracy-based greedy algorithms [Bläsius, Fischbeck; Transactions of Algorithms '24]. In an attempt to bridge this theory-practice gap we characterise the parameter of degeneracy yielding a simple approximation algorithm for colouring HRGs. The approximation ratio of our algorithm ranges from $(2/\sqrt{3})$ to $4/3$ depending on the power-law exponent of the model. We complement our findings for the degeneracy with new insights on the clique number of hyperbolic random graphs. We show that degeneracy and clique number are substantially different and derive an improved upper bound on the clique number. Additionally, we show that the core of HRGs does not constitute the largest clique. Lastly we demonstrate that the degeneracy of the closely related standard model of geometric inhomogeneous random graphs behaves inherently different compared to the one of hyperbolic random graphs.

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arXiv2025

On the Locality of Hall's Theorem

The last five years of research on distributed graph algorithms have seen huge leaps of progress, both regarding algorithmic improvements and impossibility results: new strong lower bounds have emerged for many central problems and exponential improvements over the state of the art have been achieved for the runtimes of many algorithms. Nevertheless, there are still large gaps between the best known upper and lower bounds for many important problems. The current lower bound techniques for deterministic algorithms are often tailored to obtaining a logarithmic bound and essentially cannot be used to prove lower bounds beyond $Ω(\log n)$. In contrast, the best deterministic upper bounds are often polylogarithmic, raising the fundamental question of how to resolve the gap between logarithmic lower and polylogarithmic upper bounds and finally obtain tight bounds. We develop a novel algorithm design technique aimed at closing this gap. In essence, each node finds a carefully chosen local solution in $O(\log n)$ rounds and we guarantee that this solution is consistent with the other nodes' solutions without coordination. The local solutions are based on a distributed version of Hall's theorem that may be of independent interest and motivates the title of this work. We showcase our framework by improving on the state of the art for the following fundamental problems: edge coloring, bipartite saturating matchings and hypergraph sinkless orientation. In particular, we obtain an asymptotically optimal $O(\log n)$-round algorithm for $3Δ/2$-edge coloring in bounded degree graphs. The previously best bound for the problem was $O(\log^4 n)$ rounds, obtained by plugging in the state-of-the-art maximal independent set algorithm from arXiv:2303.16043 into the $3Δ/2$-edge coloring algorithm from arXiv:1711.05469 .

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arXiv2024

Distributed Delta-Coloring under Bandwidth Limitations

We consider the problem of coloring graphs of maximum degree $Δ$ with $Δ$ colors in the distributed setting with limited bandwidth. Specifically, we give a $\mathsf{poly}\log\log n$-round randomized algorithm in the CONGEST model. This is close to the lower bound of $Ω(\log \log n)$ rounds from [Brandt et al., STOC '16], which holds also in the more powerful LOCAL model. The core of our algorithm is a reduction to several special instances of the constructive Lovász local lemma (LLL) and the $deg+1$-list coloring problem.

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arXiv2024

Distributed Lovász Local Lemma under Bandwidth Limitations

The constructive Lovász Local Lemma has become a central tool for designing efficient distributed algorithms. While it has been extensively studied in the classic LOCAL model that uses unlimited bandwidth, much less is known in the bandwidth-restricted CONGEST model. In this paper, we present bandwidth- and time-efficient algorithms for various subclasses of LLL problems, including a large class of subgraph sampling problems that are naturally formulated as LLLs. Lastly, we use our LLLs to design efficient CONGEST algorithms for coloring sparse and triangle-free graphs with few colors. These coloring algorithms are exponentially faster than previous LOCAL model algorithms.

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arXiv2024

Adaptive Massively Parallel Coloring in Sparse Graphs

Classic symmetry-breaking problems on graphs have gained a lot of attention in models of modern parallel computation. The Adaptive Massively Parallel Computation (AMPC) is a model that captures the central challenges in data center computations. Chang et al. [PODC'2019] gave an extremely fast, constant time, algorithm for the $(Δ+ 1)$-coloring problem, where $Δ$ is the maximum degree of an input graph of $n$ nodes. The algorithm works in the most restrictive low-space setting, where each machine has $n^δ$ local space for a constant $0 < δ< 1$. In this work, we study the vertex-coloring problem in sparse graphs parameterized by their arboricity $α$, a standard measure for sparsity. We give deterministic algorithms that in constant, or almost constant, time give $\text{poly} ~α$ and $O(α)$-colorings, where $α$ can be arbitrarily smaller than $Δ$. A strong and standard approach to compute arboricity-dependent colorings is through the Nash-Williams forest decomposition, which gives rise to an (acyclic) orientation of the edges such that each node has a small out-degree. Our main technical contribution is giving efficient deterministic algorithms to compute these orientations and showing how to leverage them to find colorings in low-space AMPC. A key technical challenge is that the color of a node may depend on almost all of the other nodes in the graph and these dependencies cannot be stored on a single machine. Nevertheless, our novel and careful exploration technique yields the orientation, and the arboricity-dependent coloring, with a sublinear number of adaptive queries per node.

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