Finding Cheeger cuts in hypergraphs via heat equation

Cut-matching Games for Generalized Hypergraph Ratio Cuts

“…Approx Guarantee Hypergraph cut function Main Technique Implemented CLTZ [10] Cheeger All-or-nothing SDP relaxation No Li-Milenkovic CE [30] Cheeger Submodular Clique expansion + graph spectral Yes Li-Milenkovic IPM [31] N/A Submodular Inverse power method Yes Li-Milenkovic SDP [31] Cheeger Submodular SDP relaxation No IMTY HeatEQ [21] Cheeger All-or-nothing Solving hypergraph heat equation No Louis-Makarychev [35] 𝑂 ( √︁ log 𝑛) All-or-nothing SDP relaxation No KKTY [23] 𝑂 (log 𝑛) All-or-nothing LP relaxation No This paper 𝑂 (log 𝑛) Submodular, cardinality-based Cut-matching (repeated max-flow) Yes hypergraph ratio cuts that comes with a nontrivial approximation guarantee (i.e., better than 𝑂 (𝑛) in the worst case) and also applies to generalized hypergraph cuts. Additionally, compared with approximation algorithms applying only to the standard hypergraph cut function, our method comes with a substantially faster runtime guarantee and a practical implementation.…”

“…The clique expansion method for inhomogeneous hypergraphs (CE) works by replacing each hyperedge by a weighted clique in a way that minimizes the difference between cut penalties in the clique and generalized cut penalties of the original hyperedge splitting function [30]. If w 𝑒 denotes the hyperedge splitting function for 𝑒 ∈ E and ŵ𝑒 denotes the cut penalties in the projected clique, the goal is to choose edge weights in the clique so that w 𝑒 (𝐴) ≤ ŵ𝑒 (𝐴) ≤ 𝐶w 𝑒 (𝐴) for all 𝐴 ⊆ 𝑒, (21) for the smallest possible value of 𝐶. Spectral clustering is then applied to the reduced graph. The first implementation of this method for generalized hypergraph cuts was designed for specific cut penalties that are not cardinality-based (https://github.com/ lipan00123/InHclustering, accompanying [30]).…”

“…While these methods are interesting from a theoretical perspective, they are far from practical. Another downside specifically for spectral techniques [7,10,21,30,31,55] is that their Cheeger-like approximation guarantees are 𝑂 (𝑛) in the worst case, which is true even in the graph setting. Spectral techniques also often come with approximation guarantees that get increasingly worse as the maximum hyperedge size grows [7,10,30].…”

“…Spectral techniques also often come with approximation guarantees that get increasingly worse as the maximum hyperedge size grows [7,10,30]. A limitation in another direction is that many existing hypergraph ratio cut algorithms are designed only for the simplest notion of a hypergraph cut function rather than generalized cuts [7,10,21,23,35,57]. Finally, although a number of recent techniques apply to generalized hypergraph cut functions and come with practical implementations, these do not provide theoretical approximation guarantees for global ratio cut objectives because they are either heuristics [31] or because they focus on minimizing localized ratio cut objectives that are biased to a specific region of a large hypergraph [16,19,32,46,47].…”

Extended fractional-polynomial generalizations of diffusion and Fisher-KPP equations on directed networks: Modeling neurodegenerative progression

“…The dynamics of linear Laplacian operators in directed graphs have been well studied [7][8][9][10]. In the literature, the standard graph Laplacian has been developed in a variety of ways to study spectral information, such as eigenvalues, eigenvectors, and Cheeger constants, for graphs and hypergraphs [11][12][13][14][15][16][17][18]. Aside from the extensions regarding spectral graph theory, several generalizations of the standard graph Laplacian exist particularly for exploring anomalous and nonlinear diffusion processes, most of which are restricted to undirected graphs [19][20][21][22][23][24][25][26][27][28][29][30][31], and for studying the dynamics of chemical reaction networks (CRNs) [32], which are directed networks [33,34].…”