CONGEST模型下(Δ+1)-边着色的快速确定性算法
A Fast Deterministic Algorithm for $(Δ+1)$-edge coloring in CONGEST
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中文总结 AI 辅助
针对CONGEST模型，提出首个确定性poly(Δ, log n)轮的(Δ+1)-边着色算法，其n依赖关系Õ(log⁵n)与LOCAL模型最优值一致，填补了该模型下的研究空白。
中文摘要 AI 辅助
Vizing定理表明，最大度为Δ的任何图都可使用Δ+1种颜色进行正确边着色（一般情况下这是最优的）。Bernshteyn近期的突破性成果显示，在分布式计算的LOCAL模型中，可在poly(Δ, log n)轮内确定性地找到这类(Δ+1)-边着色，其中n为输入图的顶点数[J. Comb. Theory 2022]。此后，Christiansen[STOC 2023]以及Bernshteyn与Dhawan[J. Comb. Theory, Series B, 2025]改进了该运行时间中poly(log n)部分的指数。然而，这些工作所用算法均采用大消息，这为更受限的CONGEST模型留下了高效算法的问题。我们通过提出首个CONGEST模型下(Δ+1)-边着色的poly(Δ, log n)轮算法，解答了该问题。我们的算法是确定性的，其运行时间的n依赖关系为Õ(log⁵n)，与LOCAL模型中已发表的最优依赖关系一致。
英文摘要
Vizing's theorem states that any graph of maximum degree $Δ$ can be properly edge-colored with $Δ+ 1$ colors (which is optimal in general). A recent breakthrough result by Bernshteyn showed that such a $(Δ+ 1)$-edge coloring can be found deterministically in $poly(Δ,\log n)$ rounds in the LOCAL model of distributed computing, where $n$ denotes the number of vertices of the input graph [J. Comb. Theory 2022]. Since then, the exponent in the $poly(\log n)$-part of the runtime has been improved by Christiansen [STOC 2023] and Bernshteyn and Dhawan [J. Comb. Theory, Series B, 2025]. However, the algorithms used in all of these works use large messages, leaving open the question for efficient algorithms in the more restrictive CONGEST model. We answer this question by presenting the first $poly(Δ,\log n)$-round algorithm for $(Δ+ 1)$-edge coloring in the CONGEST model. Our algorithm is deterministic and the $n$-dependency of its runtime, $\tilde{O}(\log^5 n)$, matches the best published dependency in the LOCAL model.
