发表机构
University of Michigan; School of Computer Science, Georgia Tech（密歇根大学; 佐治亚理工学院计算机学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文研究在线拼车问题，证明贪心算法达到最优立方根差异界，并推广到随机到达场景，改进了此前平方根上界。
AI 中文摘要
我们考虑在线拼车问题，其中边在线到达且必须立即定向，同时保持每个顶点的入度和出度之间的差异较小。我们证明了自然的贪心算法在T次到达后产生的差异为O(min{T^{1/3},n})。这解决了Ajtai等人提出的一个问题，他们证明了任何确定性算法必须产生Ω(min{T^{1/3},n})的差异，并给出了一种差异为O(min{T^{1/2},n})的算法。我们还在随机设置中展示了类似的从平方根到立方根的改进，其中O(n)条边从底层n顶点图G中独立采样。形式上，我们展示了对于来自任何Δ-正则图G的随机到达，O((log n)^{1/3})的界。当Δ=Ω((log n)^3)时，我们展示了更精细的界O((log n/log Δ)^{1/3}+log log n)关于差异。我们证明了前一个界中的立方根项是必要的，而log log n项已知对于来自完全图的随机到达是必要的。此前的上界为O((log n)^{1/2})，这来自Kulkarni、Reis和Rothvoss以及Aden-Ali关于在线差异的开创性工作。我们证明这种立方根型界的技术可能具有独立意义，因为先前一般界所基于的标准二次势能和次高斯分析似乎本质上无法低于平方根型保证。
英文摘要
We consider the online carpooling problem, where edges arrive online and must be oriented immediately while keeping the discrepancy between the indegree and outdegree at each vertex small. We prove that the natural Greedy algorithm incurs discrepancy $O(\min\{T^{1/3},n\})$ after $T$ arrivals. This resolves a question of Ajtai et al., who showed that any deterministic algorithm must incur $Ω(\min\{T^{1/3},n\})$ discrepancy, and gave an algorithm with $O(\min\{T^{1/2},n\})$ discrepancy. We also show a similar square-root to cube-root improvement in the stochastic setting, where $O(n)$ edges are sampled independently from an underlying $n$-vertex graph $G$. Formally, we show an $O((\log n)^{1/3})$ bound for random arrivals from any $Δ$-regular graph $G$. When $Δ= Ω((\log n)^3)$, we show the more refined bound of $O((\log n/\log Δ)^{1/3}+\log\log n)$ on the discrepancy. We show that the cube-root term in the previous bound is essential, while the $\log\log n$ term is already known to be necessary for random arrivals from complete graphs. The previous upper bounds here were $O((\log n)^{1/2})$, which follow from the breakthrough works on online discrepancy due to Kulkarni, Reis, and Rothvoss, and Aden-Ali. Our techniques for proving such cube-root-type bounds may be of independent interest, as the standard quadratic-potential and subgaussian analyses underlying the previous general bounds appear inherently unable to go below square-root-type guarantees.
Comments15 pages
