Online Algorithms for Spectral Hypergraph Sparsification

Tasuku Soma, Kam Chuen Tung, Yuichi Yoshida

Abstract: 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 \((\varepsilon , \delta )\)-spectral sparsifier with multiplicative error \(\varepsilon \) and additive error \(\delta \) that has \(O(\varepsilon ^{-2} n \log n \log r \log (1 + \varepsilon W/\delta n))\) hyperedges with high probability, where \(\varepsilon , \delta \in (0,1)\), n is the number of nodes, r is the rank of the hypergraph, and W is the sum of edge weights. The space complexity of our algorithm is \(O(n^2)\), while previous algorithms required space complexity \(\varOmega (m)\), where m is the number of hyperedges. This provides an exponential improvement in the space complexity since m can be exponential in n.