We obtain better algorithms for computing more balanced orientations and degree splits in LOCAL. Important to our result is a connection to the hypergraph sinkless orientation problem [BMNSU, SODA'25] We design an algorithm of complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log n)$ for computing a balanced orientation with discrepancy at most $\varepsilon \cdot \mathrm{deg}(v)$ for every vertex $v \in V$. This improves upon a previous result by [GHKMSU, Distrib. Comput. 2020] of complexity $\mathcal{O}(\varepsilon^{-1} \cdot \log \varepsilon^{-1} \cdot (\log \log \varepsilon^{-1})^{1.71} \cdot \log n)$. Further, we show that this result can also be extended to compute undirected degree splits with the same discrepancy and in the same runtime. As as application we show that $(3 / 2 + \varepsilon)Δ$-edge coloring can now be solved in $\mathcal{O}(\varepsilon^{-1} \cdot \log^2 Δ\cdot \log n + \varepsilon^{-2} \cdot \log n)$ rounds in LOCAL. Note that for constant $\varepsilon$ and $Δ= \mathcal{O}(2^{\log^{1/3} n})$ this runtime matches the current state-of-the-art for $(2Δ- 1)$-edge coloring in [Ghaffari & Kuhn, FOCS'21].

翻译：我们在 LOCAL 模型中获得了计算更平衡的定向和度拆分的改进算法。该结果的重要基础是与超图无源定向问题 [BMNSU, SODA'25] 的联系。我们设计了一个复杂度为 $\mathcal{O}(\varepsilon^{-1} \cdot \log n)$ 的算法，用于计算对每个顶点 $v \in V$ 的偏差至多为 $\varepsilon \cdot \mathrm{deg}(v)$ 的平衡定向。这改进了先前 [GHKMSU, Distrib. Comput. 2020] 中复杂度为 $\mathcal{O}(\varepsilon^{-1} \cdot \log \varepsilon^{-1} \cdot (\log \log \varepsilon^{-1})^{1.71} \cdot \log n)$ 的结果。此外，我们证明该结果也可扩展计算具有相同偏差和相同运行时间的无向度拆分。作为一个应用，我们表明 $(3 / 2 + \varepsilon)Δ$-边染色现在可以在 LOCAL 模型中用 $\mathcal{O}(\varepsilon^{-1} \cdot \log^2 Δ\cdot \log n + \varepsilon^{-2} \cdot \log n)$ 轮求解。注意，对于常数 $\varepsilon$ 和 $Δ= \mathcal{O}(2^{\log^{1/3} n})$，此运行时间与 [Ghaffari & Kuhn, FOCS'21] 中关于 $(2Δ- 1)$-边染色的当前最优结果相匹配。