Online Graph Balancing and the Power of Two Choices
This addresses a fundamental problem in online algorithms for load balancing, providing near-optimal performance for arbitrary graphs, which is incremental as it builds on the power-of-two choices setting.
The paper tackles the online graph balancing problem under i.i.d. arrivals from a known base graph, showing that the greedy algorithm can perform poorly with a competitive ratio of Ω(log n) for irregular graphs, but they design a new algorithm that achieves an O(log log n)-competitive ratio, which is optimal up to constants.
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.