Online Algorithms for Spectral Hypergraph Sparsification
Online Algorithms for Spectral Hypergraph Sparsification
Tasuku Soma, Kam Chuen Tung, Yuichi Yoshida
AbstractWe 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 $O(\epsilon^{-2} n (\log n)^2 \log(1 + \epsilon W/\delta n))$ 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 $n$.