Problem
Research questions and friction points this paper is trying to address.
online graph balancing
power of two choices
i.i.d. arrivals
maximum in-degree
competitive ratio
Online Graph Balancing and the Power of Two Choices
🤖 AI Summary
This work addresses the online graph orientation problem, where edges arrive sequentially and must be oriented immediately to minimize the maximum in-degree. Focusing on the i.i.d. random arrival model over an arbitrary underlying graph, the paper introduces a multiscale decomposition method based on the graph’s logarithmic skewness, partitioning the base graph into skewed biregular subgraphs. Leveraging this decomposition, a novel online algorithm is devised that overcomes the limitations of traditional greedy strategies. The proposed approach achieves a competitive ratio of $O(\log\log n)$ on any base graph, matching the known information-theoretic lower bound for the complete graph case and thus attaining optimal order.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
log-skewness
graph decomposition
online graph balancing
power of two choices
competitive analysis
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