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**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

There are a lot of recent works on generalizing the spectral theory of graphs and graph partitioning to k-uniform hypergraphs. There have been two broad directions toward this goal. One generalizes the notion of graph conductance to hypergraph conductance [Louis, Makarychev - TOC'16; Chan, Louis, Tang, Zhang - JACM'18]. In the second approach, one can view a hypergraph as a simplicial complex and study its various topological properties [Linial, Meshulam - Combinatorica'06; Meshulam, Wallach - RSA'09; Dotterrer, Kaufman, Wagner - SoCG'16; Parzanchevski, Rosenthal - RSA'17] and spectral properties [Kaufman, Mass - ITCS'17; Dinur, Kaufman - FOCS'17; Kaufman, Openheim - STOC'18; Oppenheim - DCG'18; Kaufman, Openheim - Combinatorica'20].
In this work, we attempt to bridge these two directions of study by relating the spectrum of up-down walks and swap walks on the simplicial complex, a downward closed set system, to hypergraph expansion. More precisely, we study the simplicial complex obtained by downward closing the given hypergraph and random walks between its levels X(l), i.e., the sets of cardinality l. In surprising contrast to random walks on graphs, we show that the spectral gap of swap walks and up-down walks between level m and l with 1 < m ⩽ l cannot be used to infer any bounds on hypergraph conductance. Moreover, we show that the spectral gap of swap walks between X(1) and X(k-1) cannot be used to infer any bounds on hypergraph conductance. In contrast, we give a Cheeger-like inequality relating the spectra of walks between level 1 and l for any l ⩽ k to hypergraph expansion. This is a surprising difference between swaps walks and up-down walks!
Finally, we also give a construction to show that the well-studied notion of link expansion in simplicial complexes cannot be used to bound hypergraph expansion in a Cheeger-like manner.

Anand Louis, Rameesh Paul, and Arka Ray. Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{louis_et_al:LIPIcs.SWAT.2024.33, author = {Louis, Anand and Paul, Rameesh and Ray, Arka}, title = {{Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {33:1--33:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.33}, URN = {urn:nbn:de:0030-drops-200739}, doi = {10.4230/LIPIcs.SWAT.2024.33}, annote = {Keywords: Sparse Cuts, Random Walks, Link Expansion, Hypergraph Expansion, Simplicial Complexes, High Dimensional Expander, Threshold Rank} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

The problem of finding the largest induced balanced bipartite subgraph in a given graph is NP-hard. This problem is closely related to the problem of finding the smallest Odd Cycle Transversal.
In this work, we consider the following model of instances: starting with a set of vertices V, a set S ⊆ V of k vertices is chosen and an arbitrary d-regular bipartite graph is added on it; edges between pairs of vertices in S× (V⧵S) and (V⧵S) × (V⧵S) are added with probability p. Since for d = 0, the problem reduces to recovering a planted independent set, we don't expect efficient algorithms for k = o(√n). This problem is a generalization of the planted balanced biclique problem where the bipartite graph induced on S is a complete bipartite graph; [Yevgeny Levanzov, 2018] gave an algorithm for recovering S in this problem when k = Ω(√n).
Our main result is an efficient algorithm that recovers (w.h.p.) the planted bipartite graph when k = Ω_p(√{n log n}) for a large range of parameters. Our results also hold for a natural semi-random model of instances, which involve the presence of a monotone adversary. Our proof shows that a natural SDP relaxation for the problem is integral by constructing an appropriate solution to it’s dual formulation. Our main technical contribution is a new approach for construction the dual solution where we calibrate the eigenvectors of the adjacency matrix to be the eigenvectors of the dual matrix. We believe that this approach may have applications to other recovery problems in semi-random models as well.
When k = Ω(√n), we give an algorithm for recovering S whose running time is exponential in the number of small eigenvalues in graph induced on S; this algorithm is based on subspace enumeration techniques due to the works of [Alexandra Kolla and Madhur Tulsiani, 2007; Arora et al., 2010; Kolla, 2011].

Akash Kumar, Anand Louis, and Rameesh Paul. Exact Recovery Algorithm for Planted Bipartite Graph in Semi-Random Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 84:1-84:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kumar_et_al:LIPIcs.ICALP.2022.84, author = {Kumar, Akash and Louis, Anand and Paul, Rameesh}, title = {{Exact Recovery Algorithm for Planted Bipartite Graph in Semi-Random Graphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {84:1--84:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.84}, URN = {urn:nbn:de:0030-drops-164251}, doi = {10.4230/LIPIcs.ICALP.2022.84}, annote = {Keywords: SDP duality, Planted models, Semi-random models, Exact recovery, Threshold rank, Spectral embedding, Subspace enumeration} }

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