arXiv subjects

Yann Bourreau

Publications and source records attributed to Yann Bourreau.

3 recordsLinked to original sources

arXiv2025

Faster Distributed $\Delta$-Coloring via a Reduction to MIS

Recent improvements on the deterministic complexities of fundamental graph problems in the LOCAL model of distributed computing have yielded state-of-the-art upper bounds of $\tilde{O}(\log^{5/3} n)$ rounds for maximal independent set (MIS) and $(\Delta + 1)$-coloring [Ghaffari, Grunau, FOCS'24] and $\tilde{O}(\log^{19/9} n)$ rounds for the more restrictive $\Delta$-coloring problem [Ghaffari, Kuhn, FOCS'21; Ghaffari, Grunau, FOCS'24; Bourreau, Brandt, Nolin, STOC'25]. In our work, we show that $\Delta$-coloring can be solved deterministically in $\tilde{O}(\log^{5/3} n)$ rounds as well, matching the currently best bound for $(\Delta + 1)$-coloring. We achieve our result by developing a reduction from $\Delta$-coloring to MIS that guarantees that the (asymptotic) complexity of $\Delta$-coloring is at most the complexity of MIS, unless MIS can be solved in sublogarithmic time, in which case, due to the $\Omega(\log n)$-round $\Delta$-coloring lower bound from [BFHKLRSU, STOC'16], our reduction implies a tight complexity of $\Theta(\log n)$ for $\Delta$-coloring. In particular, any improvement on the complexity of the MIS problem will yield the same improvement for the complexity of $\Delta$-coloring (up to the true complexity of $\Delta$-coloring). Our reduction yields improvements for $\Delta$-coloring in the randomized LOCAL model and when complexities are parameterized by both $n$ and $\Delta$. We obtain a randomized complexity bound of $\tilde{O}(\log^{5/3} \log n)$ rounds (improving over the state of the art of $\tilde{O}(\log^{8/3} \log n)$ rounds) on general graphs and tight complexities of $\Theta(\log n)$ and $\Theta(\log \log n)$ for the deterministic, resp.\ randomized, complexity on bounded-degree graphs. In the special case of graphs of constant clique number (which for instance include bipartite graphs), we also give a reduction to the $(\Delta+1)$-coloring problem.

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arXiv2025

Optimal Deterministic Rendezvous in Labeled Lines

In a rendezvous task, some mobile agents dispersed in a network have to gather at an arbitrary common site. We consider the rendezvous problem on the infinite labeled line, with $2$ agents, without communication, and a synchronous notion of time. Each node on the line is labeled with a unique positive integer. The initial distance between the agents is denoted by $D$. Time is divided into rounds and measured from the moment an agent first wakes up. We denote by $\tau$ the delay between the two agents' wake up times. If awake in a given round $T$, an agent at a node $v$ has three options: stay at the node $v$, take port $0$, or take port $1$. If it decides to stay, the agent will still be at node $v$ in round $T+1$. Otherwise, it will be at one of the two neighbors of $v$ on the infinite line, depending on the port it chose. The agents achieve rendezvous in $T$ rounds if they are at the same node in round $T$. We aim for a deterministic algorithm for this problem. The problem was recently considered by Miller and Pelc [Distributed Computing 2025]. With $\ell_{\max}$ the largest label of the two starting nodes, they showed that no algorithm can guarantee rendezvous in $o(D \log^* \ell_{\max})$ rounds. The lower bound follows from a connection with the LOCAL model of distributed computing, and holds even if the agents are guaranteed simultaneous wake-up ($\tau = 0$) and are told their initial distance $D$. Miller and Pelc also gave an algorithm of optimal matching complexity $O(D \log^* \ell_{\max})$ when the agents know $D$, but only obtained the higher bound of $O(D^2 (\log^* \ell_{\max})^3)$ when $D$ is unknown to the agents. We improve this complexity to a tight $O(D \log^* \ell_{\max})$. In fact, our algorithm achieves rendezvous in $O(D \log^* \ell_{\min})$ rounds, where $\ell_{\min}$ is the smallest label within distance $O(D)$ of the two starting positions.

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arXiv2025

Faster Distributed $Δ$-Coloring via Ruling Subgraphs

Brooks' theorem states that all connected graphs but odd cycles and cliques can be colored with $Δ$ colors, where $Δ$ is the maximum degree of the graph. Such colorings have been shown to admit non-trivial distributed algorithms [Panconesi and Srinivasan, Combinatorica 1995] and have been studied intensively in the distributed literature. In particular, it is known that any deterministic algorithm computing a $Δ$-coloring requires $Ω(\log n)$ rounds in the LOCAL model [Chang, Kopelowitz, and Pettie, FOCS 2016], and that this lower bound holds already on constant-degree graphs. In contrast, the best upper bound in this setting is given by an $O(\log^2 n)$-round deterministic algorithm that can be inferred already from the works of [Awerbuch, Goldberg, Luby, and Plotkin, FOCS 1989] and [Panconesi and Srinivasan, Combinatorica 1995] roughly three decades ago, raising the fundamental question about the true complexity of $Δ$-coloring in the constant-degree setting. We answer this long-standing question almost completely by providing an almost-optimal deterministic $O(\log n \log^* n)$-round algorithm for $Δ$-coloring, matching the lower bound up to a $\log^* n$-factor. Similarly, in the randomized LOCAL model, we provide an $O(\log \log n \log^* n)$-round algorithm, improving over the state-of-the-art upper bound of $O(\log^2 \log n)$ [Ghaffari, Hirvonen, Kuhn, and Maus, Distributed Computing 2021] and almost matching the $Ω(\log \log n)$-round lower bound by [BFHKLRSU, STOC 2016]. Our results make progress on several important open problems and conjectures. One key ingredient for obtaining our results is the introduction of ruling subgraph families as a novel tool for breaking symmetry between substructures of a graph, which we expect to be of independent interest.

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