类贪心缺陷着色：分布式算法与应用
Greedy-Like Defective Coloring: Distributed Algorithms and Applications
- University of Freiburg（弗赖堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文拓展了Barenboim等的两遍贪心缺陷着色算法，将其推广到列表缺陷着色问题，在CONGEST模型中得到(Δ+1)-着色的替代算法，还分析了广义算法的缺陷权衡并证明其局限性。
AI中文摘要：
图G=(V,E)的d-缺陷c-着色是指用c种颜色对节点V进行着色，使得每个节点最多有d个同色邻居。计算不同变体缺陷着色的分布式算法是当前大多数确定性分布式着色算法的核心，也是诸多其他分布式图算法的重要工具。在若干场景中，若能更高效地求解某类缺陷着色，整体复杂度可得到改善。Barenboim与Elkin[STOC '09]提出了一种两遍贪心算法，使用p²种颜色，缺陷为⌊Δ/p⌋，运行时间为O(Δ+log*n)，这仍是有界度图中O(log*n)时间算法的最优缺陷/颜色权衡。本文拓展了该两遍算法的能力：首先，将其推广到列表缺陷着色问题（Fuchs与Kuhn，[DISC '23]），进而在CONGEST模型中得到一种替代算法，可在Õ(√Δ)+O(log*n)轮内计算正确的(Δ+1)-着色；其次，分析了针对标准缺陷着色的广义两遍算法，证明若颜色数c非完全平方，多数情况下可将分布式c-着色的最优缺陷降低一个常数因子，但也存在局限性：对任意c≥1，该广义算法无法实现缺陷低于(1-o(1))·Δ/√c的c-着色。
英文摘要：
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 $\lfloor Δ/p\rfloor$ in $O(Δ+\log^{\ast} n)$ 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 $(Δ+1)$-coloring in $\tilde{O}(\sqrtΔ) + O(\log^{\ast} n)$ 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 $(1-o(1))\cdotΔ/\sqrt{c}$.
