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.
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Acknowledgements
TS is supported by JSPS KAKENHI Grant Number JP19K20212. A part of this work was done during KT’s visit to National Institute of Informatics. YY is supported by JSPS KAKENHI Grant Number JP20H05965 and JP22H05001.
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Soma, T., Tung, K.C., Yoshida, Y. (2024). Online Algorithms for Spectral Hypergraph Sparsification. In: Vygen, J., Byrka, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2024. Lecture Notes in Computer Science, vol 14679. Springer, Cham. https://doi.org/10.1007/978-3-031-59835-7_30
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