arXiv subjects

Manuel Jakob

Publications and source records attributed to Manuel Jakob.

3 recordsLinked to original sources

arXiv2026

Sublogarithmic Distributed Vertex Coloring with Optimal Number of Colors

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

cs.DS↗

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.

cs.DS↗

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].

cs.DC↗