发表机构
Weizmann Institute of Science（魏茨曼科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究 LOCAL 模型中的近均衡染色，证明精确均衡需全局时间（环上 $\Omega(n)$ 轮），而粗略均衡可局部实现，并给出匹配算法与复杂度界限。
AI 中文摘要
对于最大度为 $\Delta$ 的 $n$ 顶点图，均衡的 $(\Delta+1)$-染色是一种正常染色，其所有颜色类的大小均为 $\sigma = n/(\Delta+1)$（四舍五入）。已知的针对该目标松弛的算法沿生成结构聚合，成本随直径 $D$ 缩放。我们研究 LOCAL 模型中的近均衡染色，并证明对于确定性算法，精确均衡本质上是全局性的：即使在环 $C_n$ 上，精确均衡的自由 $3$-染色也需要 $\Omega(n)$ 轮。更一般地，加法不平衡度 $g$ 需要 $\Omega(n/g)$ 轮，与确定性 $O((n/g)\log^* n)$ 轮算法匹配（相差 $\log^* n$ 因子），并且这些问题在环上实现了稠密的中间确定性 LOCAL 复杂度族 $\tilde\Theta(n^\alpha)$，其中 $\alpha \in (0,1)$。该精确界限在另外两个意义上也是尖锐的：标识符宇宙阈值恰好是 $N=n+1$，并且 $\log^* n$ 的差距无法通过更快的统治集锚点来消除。该界限随后扩展到扭曲纤维积，这是一个显式族，对于每个 $\Delta \ge 3$，实现了从 Moore 界 $\Theta(\log n/\log\Delta)$ 到 $\Theta(n/\Delta)$ 的每个直径尺度。在该族上，精确均衡的 $(\Delta+1)$-染色确定性需要 $\Theta(D)$ 轮，因此在有界度图（对数直径）上，精确性已经需要直径时间。相比之下，粗略均衡允许局部算法。在环上，常数倍乘均衡需要 $\Theta(\log^* n)$ 轮。在具有大颜色类的一般图上，所有类的大小可以保持在平均值的 $(1\pm\eta)$ 倍以内，时间与直径无关：对于 $\Delta \le 2^{\sqrt{\log n}/C}$，调色板恰好为 $\Delta+1$；对于每个 $\Delta \le n^{1-o(1)}$，调色板为 $(1+\eta)(\Delta+1)$，后者在 $O(\log n\\,(\log\log n)^2)$ 轮内完成。
英文摘要
For an $n$-vertex graph of maximum degree $Δ$, an equitable $(Δ+1)$-coloring is a proper coloring all of whose color classes have size $σ= n/(Δ+1)$ up to rounding. Known algorithms for relaxations of this target aggregate along a spanning structure, at cost scaling with the diameter $D$. We study near-equitable coloring in the LOCAL model and prove that, for deterministic algorithms, exact balance is inherently global: already on the cycle $C_n$, exact balanced free $3$-coloring requires $Ω(n)$ rounds. More generally, additive imbalance $g$ requires $Ω(n/g)$ rounds, matching a deterministic $O((n/g)\log^* n)$-round algorithm up to the $\log^* n$ factor, and these problems realize a dense family of intermediate deterministic LOCAL complexities $\tildeΘ(n^α)$, $α\in (0,1)$, on cycles. The exact bound is sharp in two further senses: the identifier-universe threshold is exactly $N=n+1$, and the $\log^* n$ gap cannot be closed via faster ruling-set anchors. The bound then extends to twisted fiber products, an explicit family realizing, for every $Δ\ge 3$, every diameter scale from the Moore bound $Θ(\log n/\logΔ)$ up to $Θ(n/Δ)$. Exact equitable $(Δ+1)$-coloring requires $Θ(D)$ rounds deterministically on this family, so exactness costs diameter time already on bounded-degree graphs of logarithmic diameter. In contrast, coarse balance admits local algorithms. On cycles, constant multiplicative equity costs $Θ(\log^* n)$ rounds. On general graphs with large color classes, all class sizes can be kept within $(1\pmη)$ times the average in time independent of the diameter: with palette exactly $Δ+1$ for $Δ\le 2^{\sqrt{\log n}/C}$, and with palette $(1+η)(Δ+1)$ for every $Δ\le n^{1-o(1)}$, the latter in $O(\log n\,(\log\log n)^2)$ rounds.
Comments54 pages, 5 figures, 1 table
