From Graph to Hypergraph Multiway Partition: Is the Single Threshold the Only Route?

Ene, Alina; Nguyêݱn, Huy L.

doi:10.1007/978-3-662-44777-2_32

Cited by 4 publications

(6 citation statements)

References 13 publications

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“…There are also several generalizations of the standard graph multiway cut objective, including directed graph [38,97] and node-weighted variants [38,39]. The problem has also been studied in the hypergraph setting, under different hypergraph generalizations of the all-or-nothing splitting penalty [23,31,75,97]. Here we consider the goal of separating terminal nodes in order to minimize a generalized hypergraph cut function.…”

Section: Tractability Regions and Openmentioning

confidence: 99%

“…This is a special case of Gen-HyperMC where the all-or-nothing multiway splitting penalty is used. The hypergraph multiway partition problem (HyperMP) differs in that the cost at a cut hyperedge is proportional to the number of clusters spanned by the hyperedge [23,31]. This can be viewed as an instance of Gen-HyperMC when the sum of external degrees splitting function is used for all hyperedges.…”

Section: Allmentioning

confidence: 99%

“…HyperMC is approximation equivalent to the node-weighted multiway cut in graphs [75], for which a (2 − 2 k )-approximation is known [38,39]. For HyperMP, the best known approximation factor is 4 3 [31]. Both HyperMC and HyperMP are special cases of the submodular multiway partition objective (Sub-MP) [97].…”

Section: Allmentioning

confidence: 99%

“…. , k} (where k is the number of nodes in a hyperedge) is equivalent to the hypergraph multiway partition problem, which can be approximated to within a factor 4 3 [31]. Are there other regimes of cluster-based HyperMC for which we can obtain approximation guarantees?…”

Section: Allmentioning

confidence: 99%

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Hypergraph Cuts with General Splitting Functions

Veldt¹,

Benson²,

Kleinberg³

2020

Preprint

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The minimum s-t cut problem in graphs is one of the most fundamental problems in combinatorial optimization, and graph cuts underlie algorithms throughout discrete mathematics, theoretical computer science, operations research, and data science. While graphs are a standard model for pairwise relationships, hypergraphs provide the flexibility to model multi-way relationships, and are now a standard model for complex data and systems. However, when generalizing from graphs to hypergraphs, the notion of a "cut hyperedge" is less clear, as a hyperedge's nodes can be split in several ways. Here, we develop a framework for hypergraph cuts by considering the problem of separating two terminal nodes in a hypergraph in a way that minimizes a sum of penalties at split hyperedges. In our setup, different ways of splitting the same hyperedge have different penalties, and the penalty is encoded by what we call a splitting function.Our framework opens a rich space on the foundations of hypergraph cuts. We first identify a natural class of cardinality-based hyperedge splitting functions that depend only on the number of nodes on each side of the split. In this case, we show that the general hypergraph s-t cut problem can be reduced to a tractable graph s-t cut problem if and only if the splitting functions are submodular. We also identify a wide regime of non-submodular splitting functions for which the problem is NP-hard.We also analyze extensions to multiway cuts with at least three terminal nodes and identify a natural class of splitting functions for which the problem can be reduced in an approximation-preserving way to the node-weighted multiway cut problem in graphs, again subject to a submodularity property. Finally, we outline several open questions on general hypergraph cut problems.

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Section: Tractability Regions and Openmentioning

confidence: 99%

Section: Allmentioning

confidence: 99%

Section: Allmentioning

confidence: 99%

Section: Allmentioning

confidence: 99%

See 2 more Smart Citations

Hypergraph Cuts with General Splitting Functions

Veldt¹,

Benson²,

Kleinberg³

2020

Preprint

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“…[PRT12, PR12, Par13, KKL14, SKM14] studied the relation of simplicial complexes with rather different notions of Laplacian forms, and considered isoperimetric inequalities, homologies and mixing times. Ene and Nguyen [EN14] studied the hypergraph multiway partition problem (generalizing the graph multiway partition problem) and gave a 4 3 -approximation algorithm for 3-uniform hypergraphs. Concurrent to this work, [LM14b] gave approximation algorithms for hypergraph expansion, and more generally, hypergraph small set expansion; they gave an Õ k √ log napproximation algorithm and an Õ k √ OPT log r approximation bound for the problem of computing the set of vertices of size at most |V | /k in a hypergraph H = (V, E), having the least expansion.…”

Section: Related Workmentioning

confidence: 99%

Spectral Properties of Hypergraph Laplacian and Approximation Algorithms

Chan

Louis

Tang

et al. 2018

J. ACM

119

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The celebrated Cheeger's Inequality [AM85, Alo86] establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph.In this paper we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. In particular, the operator is induced by a diffusion process on the hypergraph, such that within each hyperedge, measure flows from vertices having maximum weighted measure to those having minimum. Since the operator is non-linear, we have to exploit other properties of the diffusion process to recover a spectral property concerning the "second eigenvalue" of the resulting Laplacian. Moreover, we show that higher order spectral properties cannot hold in general using the current framework.We consider a stochastic diffusion process, in which each vertex also experiences Brownian noise from outside the system. We show a relationship between the second eigenvalue and the convergence behavior of the process.We show that various hypergraph parameters like multi-way expansion and diameter can be bounded using this operator's spectral properties. Since higher order spectral properties do not hold for the Laplacian operator, we instead use the concept of procedural minimizers to consider higher order Cheeger-like inequalities. For any k ∈ N, we give a polynomial time algorithm to compute an O(log r)-approximation to the k-th procedural minimizer, where r is the maximum cardinality of a hyperedge. We show that this approximation factor is optimal under the SSE hypothesis (introduced by [RS10]) for constant values of k.Moreover, using the factor preserving reduction from vertex expansion in graphs to hypergraph expansion, we show that all our results for hypergraphs extend to vertex expansion in graphs. * A preliminary version of this paper appeared in STOC 2015 [Lou15] and the current paper is the result of a merge with [CTZ15].

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Hypergraph Markov Operators, Eigenvalues and Approximation Algorithms

Louis

2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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The celebrated Cheeger's Inequality [AM85, Alo86] establishes a bound on the expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph.In this paper we introduce a new hypergraph Laplacian operator (generalizing the Laplacian matrix of graphs) and study its spectra. We prove a Cheeger-type inequality for hypergraphs, relating the second smallest eigenvalue of this operator to the expansion of the hypergraph. We bound other hypergraph expansion parameters via higher eigenvalues of this operator. We give bounds on the diameter of the hypergraph as a function of the second smallest eigenvalue of the Laplacian operator. The Markov process underlying the Laplacian operator can be viewed as a dispersion process on the vertices of the hypergraph that can be used to model rumour spreading in networks, brownian motion, etc., and might be of independent interest. We bound the Mixing-time of this process as a function of the second smallest eigenvalue of the Laplacian operator. All these results are generalizations of the corresponding results for graphs.We show that there can be no linear operator for hypergraphs whose spectra captures hypergraph expansion in a Cheeger-like manner. Our Laplacian operator is non-linear and thus computing its eigenvalues exactly is intractable. For any k ∈ 0 , we give a polynomial time algorithm to compute an approximation to the k th smallest eigenvalue of the operator . We show that this approximation factor is optimal under the SSE hypothesis (introduced by [RS10]) for constant values of k.We give a O log k log r log log k -approximation algorithm for the general sparsest cut in hypergraphs, where k is the number of "demands" in the instance and r is the size of the largest hyperedge.Finally, using the factor preserving reduction from vertex expansion in graphs to hypergraph expansion, we show that all our results for hypergraphs extend to vertex expansion in graphs.

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From Graph to Hypergraph Multiway Partition: Is the Single Threshold the Only Route?

Cited by 4 publications

References 13 publications

Hypergraph Cuts with General Splitting Functions

Hypergraph Cuts with General Splitting Functions

Spectral Properties of Hypergraph Laplacian and Approximation Algorithms

Hypergraph Markov Operators, Eigenvalues and Approximation Algorithms

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