LOCAL模型下基于弹性Lovász局部引理的无三角图着色
Triangle-Free Coloring in LOCAL via Resilient Lovász Local Lemma
AI总结：
本研究通过弹性Lovász局部引理改进Pettie-Su无三角图着色算法，消除分布式LLL瓶颈，得到首个o(Δ)色、log^{O(1)}log n轮的无三角图着色算法，还得到匹配最优上界的围长5图着色算法。
AI中文摘要：
Lovász局部引理（LLL）是分布式算法研究中的核心概率工具，构造性LLL被证明是LOCAL模型中随机复杂度为o(log n)的局部可检验标记问题的完全问题。LLL的经典应用之一是对无三角图等稀疏结构图着色，无三角图着色因此成为亚对数随机分布式算法技术的基准问题。Pettie和Su的最优分布式无三角图着色算法（ICALP 2013、Information and Computation 2015）使用Δ/k种颜色（k最大可达(1/4−ε)lnΔ），需O(k+log*n)次分布式LLL应用，但分布式LLL本身是难题，现有最快算法需O(log_Δ n)或O(Δ/logΔ)+log^{O(1)}log n轮。本研究采用Davies（SODA 2023）提出的“弹性”定义改进Pettie-Su算法，使生成的LLL实例可在log^{O(1)}log n轮内求解，复杂度为O(k)+log^{O(1)}log n（当k=log^{ω(1)}log n时无需LLL），消除了LLL步骤的瓶颈效应。作为推论，本研究得到首个用o(Δ)种颜色对无三角图着色的log^{O(1)}log n轮算法，相同框架还生成了围长为5的图着色算法，使用(1+ε)Δ/lnΔ种颜色，复杂度为O(k)+log^{O(1)}log n，达到颜色数的最优存在上界。
英文摘要：
The Lovász Local Lemma (LLL) is a probabilistic tool that has been shown to be of central importance in the study of distributed algorithms. For example, the constructive LLL is known to be complete for the class of locally-checkable labeling problems with $o(\log n)$ randomized complexities in the LOCAL model. One classic application of the LLL is in coloring graphs with some sparse structure, such as triangle-free graphs. Triangle-free coloring therefore serves as a benchmark problem for techniques for sublogarithmic randomized distributed algorithms. The state-of-the-art distributed triangle-free coloring algorithm of Pettie and Su [ICALP 2013, Information and Computation 2015] uses $\fracΔ{k}$ colors (where $k$ can be up to $(\frac14 - \varepsilon)\ln Δ$) and consists of $O(k+\log^* n)$ applications of the distributed LLL. However, the distributed LLL is itself a difficult problem; despite significant study, the fastest algorithms known require $O(\log_Δn)$ or $O(\fracΔ{\logΔ})+\log^{O(1)}\log n$ rounds. In this work, we adapt the Pettie-Su's algorithm so that the resulting LLL instances can be solved in $\log^{O(1)}\log n$ rounds, by employing the 'resilience' definition of Davies [SODA 2023]. This gives an $O(k)+ \log^{O(1)}\log n$ complexity (since the LLL is not needed when $k= \log^{ω(1)}\log n$), essentially causing the LLL steps to no longer be the bottleneck of the algorithm. As a corollary we obtain the first $\log^{O(1)}\log n$-round algorithms for coloring triangle-free graphs with $o(Δ)$ colors. The same framework also yields a companion girth-$5$ algorithm, using $(1+\varepsilon)Δ/\ln Δ$ colors in $O(k)+ \log^{O(1)}\log n$ rounds, matching the best known existential upper bound for the number of colors.
