We provide the first online algorithm for spectral hypergraph sparsification. In the online setting, hyperedges with positive weights are arriving in a stream, and upon the arrival of each hyperedge, we must irrevocably decide whether or not to include it in the sparsifier. Our algorithm produces an $(\epsilon, \delta)$-spectral sparsifier with multiplicative error $\epsilon$ and additive error $\delta$ that has $O(\epsilon^{-2} n \log n \log r \log(1 + \epsilon W/\delta n))$ hyperedges with high probability, where $\epsilon, \delta \in (0,1)$, $n$ is the number of nodes, and $W$ is the sum of edge weights. The space complexity of our algorithm is $O(n^2)$, while previous algorithms require the space complexity of $\Omega(m)$, where $m$ is the number of hyperedges. This provides an exponential improvement in the space complexity since $m$ can be exponential in $n$.

翻译：我们提出了首个用于谱超图稀疏化的在线算法。在在线设置中，具有正权重的超边以流的形式到达，每当一条超边到达时，我们必须不可撤销地决定是否将其包含在稀疏化图中。我们的算法生成一个具有乘法误差$\epsilon$和加法误差$\delta$的$(\epsilon, \delta)$-谱稀疏化图，该图以高概率包含$O(\epsilon^{-2} n \log n \log r \log(1 + \epsilon W/\delta n))$条超边，其中$\epsilon, \delta \in (0,1)$，$n$是节点数，$W$是边权重之和。该算法的空间复杂度为$O(n^2)$，而先前算法的空间复杂度为$\Omega(m)$，其中$m$是超边数量。由于$m$可能随$n$呈指数增长，这提供了空间复杂度的指数级改进。