In the classic online graph balancing problem, edges arrive sequentially and must be oriented immediately upon arrival, to minimize the maximum in-degree. For adversarial arrivals, the natural greedy algorithm is $O(\log n)$-competitive, and this bound is the best possible for any algorithm, even with randomization. We study this problem in the i.i.d. model where a base graph $G$ is known in advance and each arrival is an independent uniformly random edge of $G$. This model generalizes the standard power-of-two choices setting, corresponding to $G = K_n$, where the greedy algorithm achieves an $O(\log\!\log n)$ guarantee. We ask whether a similar bound is possible for arbitrary base graphs. While the greedy algorithm is optimal for adversarial arrivals and also for i.i.d. arrivals from regular base graphs (such as $G = K_n$), we show that it can perform poorly in general: there exist mildly irregular graphs $G$ for which greedy is $\widetildeΩ(\log n)$-competitive under i.i.d. arrivals. In sharp contrast, our main result is an $O(\log\!\log n)$-competitive online algorithm for every base graph $G$; this is optimal up to constant factors, since an $Ω(\log\!\log n)$ lower bound already holds even for the complete graph $G = K_n$. The key new idea is a notion of log-skewness for graphs, which captures the irregular substructures in $G$ that force the offline optimum to be large. Moreover, we show that any base graph can be decomposed into ``skew-biregular'' pieces at only $O(\log\!\log n)$ scales of log-skewness, and use this to design a decomposition-based variant of greedy that is $O(\log\!\log n)$-competitive.

翻译：在经典在线图均衡问题中，边按序到达并须在到达时立即定向，以最小化最大入度。对于对抗性到达场景，自然贪心算法具有$O(\log n)$竞争比，且该界对任何算法（包括随机化算法）均为最优。我们研究独立同分布模型下的该问题：已知基图$G$，每次到达是$G$中独立均匀随机的一条边。该模型推广了对应$G = K_n$的标准双选择设置，此时贪心算法能达到$O(\log\!\log n)$保证。我们探究任意基图是否可实现类似界值。尽管贪心算法在对抗性到达及正则基图（如$G = K_n$）的独立同分布到达中均为最优，但研究表明它在一般情况下表现欠佳：存在轻度非正则图$G$，使得贪心算法在独立同分布到达下具有$\widetildeΩ(\log n)$竞争比。与此形成鲜明对比的是，我们的主要成果是为任意基图$G$设计出$O(\log\!\log n)$竞争比的在线算法；由于完全图$G = K_n$已存在$Ω(\log\!\log n)$下界，该界在常数因子意义上最优。关键创新在于引入图的偏斜度概念，该概念刻画了迫使离线最优值增大的$G$中非正则子结构。进一步，我们证明任意基图可分解为在$O(\log\!\log n)$个偏斜度尺度上的"偏斜-双正则"片段，并据此设计基于分解的贪心变体算法，实现$O(\log\!\log n)$竞争比。