Online Algorithms for Spectral Hypergraph Sparsification

TL;DR

This paper introduces the first online algorithm for spectral hypergraph sparsification, efficiently maintaining a sparse approximation of a hypergraph as edges arrive, with significantly reduced space complexity.

Contribution

The authors develop the first online spectral hypergraph sparsification algorithm with exponential space complexity improvement over previous methods.

Findings

Produces an $(psilon, elta)$-spectral sparsifier with $O(psilon^{-2} n \u2206 n \u2206 r \u2206 psilon W/elta n)$ hyperedges.

Achieves $O(n^2)$ space complexity, exponentially better than previous $\u2206(m)$ algorithms.

Provides high-probability guarantees on the quality and size of the sparsifier.

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 $(\epsilon, \delta)$-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 $n$.