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Recently, a number of variants of the notion of cut-preserving hypergraph sparsification have been studied in the literature. These variants include directed hypergraph sparsification, submodular hypergraph sparsification, general notions of approximation including spectral approximations, and more general notions like sketching that can answer cut queries using more general data structures than just sparsifiers. In this work, we provide reductions between these different variants of hypergraph sparsification and establish new upper and lower bounds on the space complexity of preserving their cuts. Specifically, we show that: 1) (1 ± ε) directed hypergraph spectral (respectively cut) sparsification on n vertices efficiently reduces to (1 ± ε) undirected hypergraph spectral (respectively cut) sparsification on n² + 1 vertices. Using the work of Lee and Jambulapati, Liu, and Sidford (STOC 2023) this gives us directed hypergraph spectral sparsifiers with O(n² log²(n) / ε²) hyperedges and directed hypergraph cut sparsifiers with O(n² log(n)/ ε²) hyperedges by using the work of Chen, Khanna, and Nagda (FOCS 2020), both of which improve upon the work of Oko, Sakaue, and Tanigawa (ICALP 2023). 2) Any cut sketching scheme which preserves all cuts in any directed hypergraph on n vertices to a (1 ± ε) factor (for ε = 1/(2^{O(√{log(n)})})) must have worst-case bit complexity n^{3 - o(1)}. Because directed hypergraphs are a subclass of submodular hypergraphs, this also shows a worst-case sketching lower bound of n^{3 - o(1)} bits for sketching cuts in general submodular hypergraphs. 3) (1 ± ε) monotone submodular hypergraph cut sparsification on n vertices efficiently reduces to (1 ± ε) symmetric submodular hypergraph sparsification on n+1 vertices. Using the work of Jambulapati et. al. (FOCS 2023) this gives us monotone submodular hypergraph sparsifiers with Õ(n / ε²) hyperedges, improving on the O(n³ / ε²) hyperedge bound of Kenneth and Krauthgamer (arxiv 2023). At a high level, our results use the same general principle, namely, by showing that cuts in one class of hypergraphs can be simulated by cuts in a simpler class of hypergraphs, we can leverage sparsification results for the simpler class of hypergraphs.
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This content will become publicly available on August 11, 2026
On the Spectral Expansion of Monotone Subsets of the Hypercube
We study the spectral gap of subgraphs of the hypercube induced by monotone subsets of vertices. For a monotone subset A ⊆ {0, 1}n of density μ(A), the previous best lower bound on the spectral gap, due to Cohen [Coh16], was γ ≳ μ(A)/n2, improving upon the earlier bound γ ≳ μ(A)2/n2 established by Ding and Mossel [DM14]. In this paper, we prove the optimal lower bound γ ≳ μ(A)/n. As a corollary, we improve the mixing time upper bound of the random walk on constant-density monotone sets from O(n3), as shown by Ding and Mossel, to O(n2). Along the way, we develop two new inequalities that may be of independent interest: (1) a directed L2-Poincar´e inequality on the hypercube, and (2) an “approximate” FKG inequality for monotone sets
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- Award ID(s):
- 2022448
- PAR ID:
- 10631078
- Publisher / Repository:
- International Conference on Randomization and Computation (RANDOM
- Date Published:
- Format(s):
- Medium: X
- Location:
- Berkeley, California
- Sponsoring Org:
- National Science Foundation
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Recently, a number of variants of the notion of cut-preserving hypergraph sparsification have been studied in the literature. These variants include directed hypergraph sparsification, submodular hypergraph sparsification, general notions of approximation including spectral approximations, and more general notions like sketching that can answer cut queries using more general data structures than just sparsifiers. In this work, we provide reductions between these different variants of hypergraph sparsification and establish new upper and lower bounds on the space complexity of preserving their cuts. Specifically, we show that: 1) (1 ± ε) directed hypergraph spectral (respectively cut) sparsification on n vertices efficiently reduces to (1 ± ε) undirected hypergraph spectral (respectively cut) sparsification on n² + 1 vertices. Using the work of Lee and Jambulapati, Liu, and Sidford (STOC 2023) this gives us directed hypergraph spectral sparsifiers with O(n² log²(n) / ε²) hyperedges and directed hypergraph cut sparsifiers with O(n² log(n)/ ε²) hyperedges by using the work of Chen, Khanna, and Nagda (FOCS 2020), both of which improve upon the work of Oko, Sakaue, and Tanigawa (ICALP 2023). 2) Any cut sketching scheme which preserves all cuts in any directed hypergraph on n vertices to a (1 ± ε) factor (for ε = 1/(2^{O(√{log(n)})})) must have worst-case bit complexity n^{3 - o(1)}. Because directed hypergraphs are a subclass of submodular hypergraphs, this also shows a worst-case sketching lower bound of n^{3 - o(1)} bits for sketching cuts in general submodular hypergraphs. 3) (1 ± ε) monotone submodular hypergraph cut sparsification on n vertices efficiently reduces to (1 ± ε) symmetric submodular hypergraph sparsification on n+1 vertices. Using the work of Jambulapati et. al. (FOCS 2023) this gives us monotone submodular hypergraph sparsifiers with Õ(n / ε²) hyperedges, improving on the O(n³ / ε²) hyperedge bound of Kenneth and Krauthgamer (arxiv 2023). At a high level, our results use the same general principle, namely, by showing that cuts in one class of hypergraphs can be simulated by cuts in a simpler class of hypergraphs, we can leverage sparsification results for the simpler class of hypergraphs.more » « less
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