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Spectral Properties of Hypergraph Laplacian and Approximation Algorithms

T-H. Hubert Chan, Anand Louis, Zhihao Gavin Tang, Chenzi Zhang

arXiv:1605.01483v1cs.DM

TL;DR

Hypergraph spectra have lacked a suitable model for estimating combinatorial parameters, and graph replacements can distort those parameters when edges are large. The paper defines a diffusion-induced hypergraph Laplacian and relates its spectral properties to hypergraph expansion; for regular hypergraphs, the stated mixing-time bound is O(log n).

  • Problem

    A spectral model for hypergraphs that estimates combinatorial parameters remained open, while replacing large hyperedges with graph structures can separate combinatorial parameters by Ω(r).

    Source: §1 · pp. 3–4

  • Method

    The paper defines a normalized hypergraph Laplacian induced by a diffusion process and generalized from the graph random-walk operator.

    Source: §1 · p. 4, §3.1 · p. 9

  • Results

    For regular hypergraphs, Theorem 3.4 says the mixing time can be O(log n).

    Source: §3.2 · p. 10

  • Takeaways & Limitations

    The operator's bounds generalize corresponding graph bounds, with most independent of maximum hyperedge size and none polynomial in that size.

    Source: §1 · p. 4

    The paper states that no bounds have polynomial dependence on maximum edge size r, although some have logarithmic dependence.

    Source: §3.6.1 · p. 16

Abstract

from arXiv · show

The celebrated Cheeger's Inequality 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 positive integer $k$, 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 for constant values of $k$.

1 Introduction

Graph spectra support Cheeger-style estimates of expansion, but a useful spectral model for hypergraphs remained open; existing tensor and graph-replacement approaches have limitations. The paper introduces a hypergraph Laplacian generalized from the graph random walk, with bounds that are mostly independent of maximum edge size.

1 Introduction

  • A spectral model could connect hypergraph spectra to parameters of practical interest, while graph spectral partitioning is already used for efficient, high-quality solutions.Hypergraph expansion also has applications in parallel and distributed computing, VLSI design, computer architecture, and scientific computing.Source: §1 · p. 3
  • Tensor-based models are also problematic: optimization is NP-hard, good approximation may be intractable, and tensor spectral properties seem unrelated to hypergraph combinatorics.The paper notes these concerns alongside the lack of a canonical matrix representation for hypergraphs.Source: §1 · pp. 3–4
  • Replacing each hyperedge by a clique or low-degree expander can separate hypergraph and graph combinatorial parameters by Ω(r), making this approach unsuitable when r is large.Here r is the largest hyperedge size.Source: §1 · p. 4
  • The paper introduces a hypergraph Laplacian that generalizes the graph random-walk operator and does not require uniform hyperedges.Most resulting bounds are independent of r; some depend logarithmically on r, and none depend polynomially on r.Source: §1 · p. 4

1.1 Related Work

  • Earlier work studied canonical tensors, clique replacements, characteristic-polynomial roots, and connections between tensor eigenvectors and hypergraph structure.The cited work includes Freidman and Wigderson, Rodriguez, Cooper and Dutle,, and Lenz and Mubayi [LM12] [LM13].Source: §1.1 · p. 4
  • Related work studied hypergraph multiway partitioning, graph vertex expansion, a tug-of-war Laplacian operator on graphs, and higher graph-Laplacian eigenvalues related to graph multi-partitioning parameters.Source: §1.1 · pp. 4–5

2 Notation

This section establishes the weighted hypergraph notation and the spaces used to express the paper's operators and diffusion. It also defines hop-diameter and relates edge expansion to the discrepancy ratio.

2 Notation

  • The hypergraph is edge-weighted; vertex weights are sums of incident edge weights, and r denotes the largest hyperedge size.The notation also distinguishes the smallest hyperedge size, regularity, and uniformity.Source: §2 · p. 5
  • Hop-diameter is the minimum maximum path length needed to connect every pair of vertices, with paths formed by intersecting consecutive hyperedges.Source: §2 · p. 5
  • The discrepancy ratio extends the expansion viewpoint from indicator vectors: for a set indicator, it equals that set's edge expansion.Source: §2 · p. 5
  • Weighted, normalized, and measure spaces encode the same vectors in different forms; operators transfer between these spaces while preserving eigenvalues.The measure-space representation is used for the diffusion process and can contain negative coordinates.Source: §2 · pp. 5–6

3 Overview of Results

The paper defines a hypergraph Laplacian through diffusion and relates its spectral properties to hypergraph expansion. The construction yields a useful second-eigenvalue result, but the current framework does not extend generally to higher-order eigenvalues.

3 Overview of Results

  • The hypergraph Laplacian is defined through a diffusion process, with spectral properties related to the expansion of the underlying hypergraph.The operator generalizes the random-walk operator on graphs.Source: §3 · p. 6, §3.1 · p. 6

3.1 Laplacian and Diffusion Process

  • For graphs, the diffusion moves measure in the direction Mϕ − ϕ, yielding dϕ/dt = −Lϕ for L = I − M.Here M = AW−1 is the random-walk transition matrix applied to the measure vector.Source: §3.1 · pp. 6–7
  • Within each hyperedge, measure flows from vertices with maximum weighted measure to those with minimum, according to the diffusion rules.The framework specifies the total flow rate for each edge and then distributes it among eligible vertex pairs.Source: §3.1 · p. 7
  • Assigning an edge’s flow to one pair or distributing it evenly can violate the rule that flow must continue between the current maximum and minimum vertices.The examples show that unequal losses from other edges can immediately change which vertices are maxima.Source: §3.1 · p. 8
  • Unlike the graph case, where procedural minimizers give the eigenvalues, hypergraph γ2 is uniquely defined but γ3 can depend on the choice of minimizer x2.For graphs, the sequence of procedural minimizers gives the Laplacian eigenvalues.Source: §3.1 · p. 9
  • The diffusion induces a normalized Laplacian whose Rayleigh quotient equals the discrepancy ratio, and a procedural minimizer attaining γ2 is an eigenvector.The theorem also implies non-negative eigenvalues.Source: §3.1 · p. 9
  • Example B.4 shows that a procedural minimizer attaining γ3 need not be an eigenvector of the Laplacian projected orthogonally to x1 and x2.The authors say this suggests the current diffusion framework cannot generally be extended to higher-order eigenvalues; whether other eigenvalues exist is left open.Source: §3.1 · p. 9
  • The averaging operator captures a sum of pairwise squared differences rather than the maximum discrepancy, leaving an Ω(r) gap between expansion and the square root of its second eigenvalue.Source: §3.1 · p. 9

3.2 Diffusion Processes

The paper studies deterministic and stochastic hypergraph diffusion, relating the spectral gap to convergence and combinatorial properties. It also identifies limits on computing a slowly mixing start and on using higher-order spectral properties.

Diffusion model

  • With no external noise, the diffusion is a deterministic process in a closed system, while the stochastic version adds Brownian noise to vertex measures.The noise rate is proportional to each vertex’s weight; setting η = 0 recovers deterministic diffusion.Source: §3.2 · p. 10

Stochastic convergence

  • For γ₂ = min_{x≠0, x⊥x₁} R(x), at each t ≥ 0, ∥ΠX_t∥₂ is stochastically dominated by ∥bX_t∥₂, whose distribution includes e^-γ₂tΠX₀ plus a Gaussian term.Source: §3.2 · p. 10

Mixing time

  • For regular hypergraphs, the deterministic diffusion’s mixing time can be O(log n).The stated regularity condition is that all vertex weights are equal.Source: §3.2 · p. 10

Local clustering

  • If diffusion remains far from stationarity through time T, the hypergraph contains a set with small expansion.The theorem’s condition compares the initial and time-T distributions with the stationary distribution.Source: §3.2 · pp. 10–11

Mixing time

  • A sparse cut guarantees some slowly mixing initial distribution, but the paper leaves open whether such a distribution can be found in polynomial time.The existence result requires an initial distribution sufficiently far from stationarity.Source: §3.2 · p. 11

Diameter

  • The paper also uses a discretized diffusion operator to prove an upper bound on hypergraph hop-diameter.The supplied theorem statement does not specify the bound’s formula.Source: §3.2 · p. 11

3.3 Cheeger Inequalities

The paper extends Cheeger inequalities to hypergraphs, using the Laplacian’s second eigenvalue for the basic result. For higher orders, it turns to orthogonal minimaximizers because the Laplacian’s higher-order spectral properties are unavailable.

Basic Cheeger inequality

  • The basic hypergraph Cheeger inequality relates hypergraph expansion to the Laplacian’s second eigenvalue γ2.The paper defines expansion for an edge-weighted hypergraph.Source: §3.3 · p. 11

Higher-order inequalities

  • Higher-order inequalities use orthogonal minimaximizers rather than higher-order spectral properties of the Laplacian.The paper notes that the sequence of procedural minimizers need not be unique.Source: §3.3 · p. 11

Orthogonal minimaximizers

  • For 2-graphs, ξk, γk, and ζk coincide with Laplacian eigenvalues, while generally ξk ≤ γk ≤ ζk ≤ kξk.In particular, γ2 = ζ2, though ξ2 can be smaller than γ2.Source: §3.3 · p. 12

Higher-order inequalities

  • Under k orthonormal vectors with discrepancy at most ξ, the higher-order results give a small set of size O(|V|/k) and Θ(k) disjoint nonempty sets.The statements cover small-set expansion and higher-order Cheeger inequalities, respectively.Source: §3.3 · p. 12

3.4 Hardness via Vertex Expansion in 2-Graphs

The paper connects graph vertex expansion to hypergraph expansion through a reduction, then uses that connection to transfer approximation and hardness results. The resulting bounds depend on degree or maximum hyperedge size.

3.4 Hardness via Vertex Expansion in 2-Graphs

  • Vertex expansion is an algorithmic primitive and a tool for proving communication lower bounds.The section describes it through the Poincaré-type parameter λ∞.Source: §3.4 · p. 13

Reduction 3.14

  • Reduction 3.14 creates one unit-weighted hyperedge from each graph vertex and its external neighborhood.For graphs with maximum degree d, the resulting hyperedges have cardinality at most d + 1 under the stated minimum-degree condition.Source: Reduction 3.14 · pp. 13–14
  • The reduction relates λ∞ on a d-regular graph to γ2 on its constructed hypergraph, enabling transfer of hypergraph expansion results to vertex expansion.The two expansion measures use different normalizations: boundary vertices over set size versus crossing hyperedges over volume.Source: Reduction 3.14 · p. 14
  • For maximum degree d, λ∞ has an O(log d)-approximation, and SSE-hardness rules out better-than-constant-times-log-d approximation; approximating γ2 within O(log r) is also SSE-hard.Here r is the largest hyperedge size; the hardness statement for γ2 concerns outputting an upper estimate within that factor.Source: Reduction 3.14 · p. 14

3.5 Approximation Algorithms

The section develops a randomized polynomial-time approximation for procedural minimizers and applies it to hypergraph expansion problems. Its central guarantee approximates ξk within O(k log r), where r is the largest hyperedge size.

Approximate Procedural Minimizers

  • O(k log r)-approximation to ξk is achieved by a randomized polynomial-time algorithm that outputs k orthonormal vectors with high probability.The algorithm builds on a procedure that finds a vector orthogonal to previously chosen vectors with discrepancy O(γ log r).Source: §3.5 · pp. 14–15

Algorithmic Applications

  • Theorem 3.21 yields randomized polynomial-time approximation algorithms for hyperedge expansion, small set expansion, and multi-way hyperedge expansion.The section introduces these applications after noting that exact orthonormal vectors attaining ξk are not known to be efficiently findable.Source: §3.5 · pp. 14–15
  • Corollary 3.22 gives a randomized polynomial-time algorithm that outputs a set with a hyperedge-expansion bound involving the largest hyperedge size r.The supplied theorem text does not preserve the complete bound.Source: §3.5 · p. 15
  • Corollary 3.23 gives a randomized polynomial-time small-set-expansion algorithm whose output size is bounded with high probability.The supplied passage fragments do not preserve the complete bound.Source: §3.5 · p. 15
  • Corollary 3.24 gives a randomized polynomial-time algorithm that produces Θ(k) non-empty disjoint sets for multi-way hyperedge expansion with high probability.The supplied passage fragments do not preserve the complete expansion bound.Source: §3.5 · p. 15

3.6 Sparsest Cut with General Demands

For hypergraph sparsest cut with general demands, the section formulates the objective over demand pairs and gives a randomized polynomial-time algorithm. The discussion emphasizes that the paper's bounds avoid polynomial dependence on the largest hyperedge size.

3.6 Sparsest Cut with General Demands

  • The general-demands objective is to find a vertex subset minimizing Φ(S), the expansion with respect to the demand pairs.The instance consists of a weighted hypergraph and demand pairs with specified demands.Source: §3.6 · p. 15
  • Theorem 3.25 gives a randomized polynomial-time algorithm for hypergraph sparsest cut with k demand pairs, returning a set with high probability.Its stated guarantee includes log k, log r, and log log k.Source: §3.6 · p. 16

3.6.1 Discussion

  • The paper stresses that none of its bounds depend polynomially on r, the largest hyperedge size, and that its results generalize the corresponding 2-graph results.It notes that polynomial dependence on r would be of little practical utility for typical instances with r = Θ(n^α), α = Ω(1).Source: §3.6.1 · p. 16

3.7 Organization

  • The paper's organization proceeds from defining diffusion and Laplacian operators to stochastic diffusion, Cheeger inequalities, hardness results, and approximation algorithms.The listed later sections include procedural-minimizer approximation and sparsest cut with general demands.Source: §3.7 · p. 16

4 Defining Diffusion Process and Laplacian for Hypergraphs

This section defines a hypergraph Laplacian through a diffusion process, using an LP characterization of densest subsets to make diffusion rates well-defined and relate the discrepancy ratio to a Rayleigh quotient. It also establishes local continuity of the diffusion and projection properties of the normalized Laplacian.

4 Defining Diffusion Process and Laplacian for Hypergraphs

  • The normalized Laplacian’s second parameter γ2 is an eigenvalue, with any minimizer of the discrepancy ratio serving as a corresponding eigenvector.The discrepancy ratio coincides with the associated Rayleigh quotient.Source: §4 · p. 16
  • The Laplacian’s second-eigenvalue result does not extend to γ3, as shown by Example B.4.Source: §4 · p. 16
  • For each hyperedge, measure flows only from maximum weighted-measure vertices to minimum ones, at a total rate equal to the edge weight times their discrepancy.The resulting operator’s Rayleigh quotient equals its discrepancy ratio in both weighted and normalized spaces, regardless of how flow weight is distributed among eligible vertex pairs.Source: §4 · p. 17

4.1 Defining Diffusion Process to Construct Laplacian

  • The diffusion constraints determine a unique derivative vector even when the induced symmetric matrix is not unique, and the resulting operator is positively homogeneous.For α > 0, Lw(αg) = αLw(g).Source: §4.1 · p. 18

4.2 A Densest Subset Problem

  • Every nonempty level set of an optimal solution to the densest-subset LP is a densest subset, with density equal to the LP objective value.The problem uses a hypergraph with possibly repeated hyperedges and seeks a nonempty subset maximizing δ(P) = (c(IP) − c(SP))/w(P).Source: §4.2 · pp. 19–20
  • Densest subsets are closed under union and nonempty intersection, and the unique maximal densest subset contains all densest subsets.Source: §4.2 · pp. 20–21
  • The diffusion-vector procedure returns the same derivative vector regardless of which densest subset it selects, assigning each vertex in a maximal densest subset the value δ(P).The remaining instance has density at most δ(X), with equality exactly when adjoining its subset preserves the original density.Source: §4.2 · p. 21
  • The procedure returns the same rates regardless of the densest subset chosen; it assigns the maximal densest subset’s vertices their density, with strictly lower maximum density afterward.This is stated for the unique maximal densest subset P after it is removed from the instance.Source: §4.2 · p. 21

4.3 Densest Subset Procedure Defines Laplacian

  • Rules (R1)–(R3) uniquely determine the diffusion rate vector, inducing L_w f = −r and Lϕ = −Wr.The rates can be found by the procedure in Figure 4.1.Source: §4.3 · p. 21
  • At any time, the diffusion rate is locally continuous, and splitting an equal-value class cannot cause a sudden rate jump.If a subset splits with equal density, the remaining vertices retain that density in the reduced instance, preserving the rate assigned by the procedure.Source: §4.3 · p. 23

4.4 Spectral Properties of Laplacian

  • The normalized Laplacian maps vectors orthogonally to x1, preserves the quadratic form under projection orthogonal to x1, and ignores the x1 component of its input.Specifically, L(x) ⊥ x1, ⟨x, Lx⟩ = ⟨Πx, LΠx⟩, and L(αx1 + βx) = βL(x).Source: §4.4 · p. 24

5 Diffusion Processes

This section develops stochastic and discretized variants of hypergraph diffusion and uses spectral properties to relate diffusion behavior to hypergraph structure, including convergence, bottlenecks, mixing time, and hop-diameter. For the diameter bound, discretized diffusion spreads a one-vertex support until every pair of vertices has positive support.

5.1 Stochastic Diffusion Process

  • The stochastic process adds independent Brownian noise at each vertex with variance proportional to its weight; setting η = 0 recovers closed-system diffusion.The section also gives an upper bound on mixing time for the deterministic η = 0 process.Source: §5.1 · pp. 25, 27
  • The transient deviation in stochastic diffusion is stochastically dominated by a Gaussian comparator whose initial component decays at rate γ2, linking larger γ2 to greater proximity to stationarity.The comparison accounts for continuing noise, so the deviations need not vanish; their distributions are bounded in mean and variance as time tends to infinity.Source: §5.1 · pp. 25–26

5.2 Bottlenecks for the Hypergraph Diffusion Process

  • If a diffusion process starts more than δ from stationarity and remains more than δ away at time T, the hypergraph contains a set with small expansion.The theorem’s set is also constrained by its stationary mass.Source: §5.2 · pp. 27–28

5.3 Lower Bounds on Mixing Time

  • For any nonzero vector y in the normalized space with y ⊥ x1 and R(y)=γ, starting diffusion at y gives, for all t≥0, ∥yt∥2 ≥ e−γt · ∥y0∥2.Source: §5.3 · p. 28

5.4 Hypergraph Diameter

  • The section defines a discretized diffusion operator and relates hypergraph hop-diameter to vertex weights and γ2, the normalized Laplacian’s second eigenvalue.The diameter result is stated in terms of the hypergraph’s vertex weights and γ2.Source: §5.4 · p. 29
  • Applying M spreads measure from a support S to vertices one hyperedge-hop from S, so iterating M tracks support growth across the hypergraph.The proof uses this mechanism to relate diffusion to hop-distance.Source: §5.4 · p. 30
  • To show hop-diameter at most l, it suffices for l iterations of M from each single-vertex measure to reach all vertices; in normalized space, the target is ⟨χu, M^l(χv)⟩ > 0 for every u and v.The normalized-space formulation uses characteristic unit vectors for vertices.Source: §5.4 · p. 30
  • The normalized calculation projects orthogonally to x1, and M(x) ⊥ x1 while M(αx1 + x) = αx1 + M(x).The stated identities hold for all x and real α.Source: §5.4 · p. 30
  • The ensuing estimate uses Cauchy–Schwarz and applies Lemma 5.11 l times.The supplied passage does not include the full inequality or its resulting bound.Source: §5.4 · p. 30

6 Cheeger Inequalities for Hypergraphs

This section develops Cheeger-style links between hypergraph expansion and Laplacian or weighted 2-graph constructions, including higher-order results. It presents algorithms and arguments for finding small-expansion sets and disjoint families whose subsets have controlled expansion.

6.1 Basic Cheeger Inequalities for Hypergraphs

  • The basic hypergraph Cheeger inequalities relate expansion φ_H to the Laplacian eigenvalue γ_2.The supplied theorem passages identify these quantities but do not preserve the inequality’s formula.Source: §6.1 · p. 31

6.2 Higher Order Orthogonal Minimaximizers

  • Because higher-order spectral properties of L are unknown, the section uses orthogonal minimaximizers based on discrepancy ratios and compares γ_k, ξ_k, and ζ_k.One stated comparison is ξ_k ≤ γ_k; the authors also report no efficient method for finding vectors attaining ξ_k or ζ_k.Source: §6.2 · p. 34

6.3 Small Set Expansion

  • Given k orthonormal weighted-space vectors with discrepancy ratios at most ξ, a randomized polynomial-time algorithm constructs a set satisfying the theorem’s small-set bounds with Ω(1) probability.The supplied theorem passages do not preserve the exact size and expansion bounds; r denotes the largest hyperedge size.Source: §6.3 · p. 36
  • Algorithm 1 produces a set of size O(n/k) with expansion bounded using the spectral parameter D, with the proof establishing the guarantee with positive probability.The supplied passages omit parts of the expansion bound and probability value.Source: §6.3 · pp. 40–41
  • Replacing each hyperedge by a weighted clique yields a 2-graph whose cut numerator is at least the hypergraph cut numerator, so the resulting set satisfies φH(S) ≤ φG(S).Each pair in a hyperedge e receives weight we/(re − 1); the construction preserves vertex weights and the discrepancy comparison uses.Source: §6.3 · p. 41

6.4 Higher Order Cheeger Inequalities for Hypergraphs

  • Given k orthonormal vectors with max_s Dw(fs) ≤ ξ, a randomized polynomial-time procedure returns ⌊(1 − ε)k⌋ non-empty disjoint sets with small expansion, with Ω(1) probability.The guarantee applies for ε ≥ 1/k.Source: §6.4 · p. 41
  • With probability at least 3/4, every vertex lies in at least one S′_l; in that event, Algorithm 2 returns at least floor(k(1 − ε)) non-empty disjoint subsets.Source: §6.4 · p. 43
  • The algorithm produces at least t ≥ (1−ε)k disjoint subsets B_l, each satisfying a bound on ν(B_l).The supplied passage truncates the bound's formula.Source: §6.4 · p. 45
  • Each such B_l contains a nonempty subset whose expansion φ is bounded.This follows from Lemma 6.17; the supplied passage truncates the bound's formula.Source: §6.4 · p. 45

7 Vertex Expansion in 2-Graphs and Hardness

The section relates the hypergraph parameter γ2 to λ∞ for vertex expansion in regular 2-graphs, then establishes SSE-hardness results for hypergraph expansion under stated edge-size conditions.

7 Vertex Expansion in 2-Graphs and Hardness

  • The reduction from vertex expansion in d-regular 2-graphs relates λ∞ to the hypergraph Laplacian parameter γ2.The section applies hypergraph edge-expansion results to vertex expansion through this reduction.Source: §7 · p. 45

7.1 Hardness via the Small-Set Expansion Hypothesis

  • Reduction 3.14 transfers the SSE-hardness of vertex expansion to hyperedge expansion.The proof applies the reduction to a graph instance and uses the resulting hypergraph’s maximum edge cardinality r = d + 1.Source: §7.1 · pp. 46–47
  • Theorem 7.3 establishes SSE-hardness under conditions on bεr log r and the minimum hyperedge size rmin.The stated conditions include bεr log r ∈ [η², c2] and rmin ≥ c1r.Source: §7.1 · p. 46
  • Theorem 7.4 states an SSE-hard distinction for hypergraphs with maximum edge size r, subject to additional parameter conditions.The stated conditions include εr log r ∈ [η², c2], rmin ≥ c1r, and a bound on γ2.Source: §7.1 · p. 47
  • In the NO case of Theorem 7.3, γ2 = Ω(bε log r).The proof uses the hypergraph Cheeger inequality alongside the hardness result.Source: §7.1 · p. 47

8 Polynomial Time Approximation Algorithm for Procedural Minimizers

The section gives a randomized polynomial-time algorithm for approximating procedural minimizers, using an SDP and rounding to construct orthogonal vectors with bounded discrepancy. These vectors can then be used by procedures for downstream hypergraph approximation algorithms.

8 Polynomial Time Approximation Algorithm for Procedural Minimizers

  • The randomized polynomial-time algorithm produces k orthonormal vectors with maximum discrepancy ratio at most k log r · ξk.The resulting vectors serve as inputs to procedures for hypergraph expansion, small-set expansion, and diameter approximation.Source: §8 · p. 48
  • For each previously constructed orthonormal set, a randomized polynomial-time procedure returns a non-zero orthogonal vector f with Dw(f) = O(γ log r).Here r is the largest hyperedge size.Source: §8 · p. 48

8.1 An SDP Relaxation to Approximate Procedural Minimizers: Proof of Theorem 8.2

  • The SDP minimizes discrepancy while enforcing orthogonality after rounding, yielding an O(log r)-approximation with constant probability.The SDP represents each vertex coordinate with a vector; the rounding procedure samples a Gaussian vector to obtain f.Source: §8.1 · p. 48, SDP 8.3 · p. 49

SDP 8.3

  • The rounding algorithm outputs a non-zero vector orthogonal to the previous vectors with probability 1, when feasibility holds.Its approximation guarantee is Dw(f) ≤ 384 log r · SDPval with probability at least 1/24.Source: SDP 8.3 · p. 49

9 Sparsest Cut with General Demands

For sparsest cut with general demands, the section presents a randomized polynomial-time approximation algorithm based on an SDP and rounding. Its stated guarantee depends on the number of demand pairs and maximum hyperedge size.

9 Sparsest Cut with General Demands

  • The randomized polynomial-time algorithm gives a sparsest-cut approximation whose stated factor is log k log r log log k.The input has k demand pairs in the hypergraph.Source: §9 · p. 51

SDP 9.2

  • The SDP uses vertex vectors and an edge objective based on the maximum squared distance within each hyperedge, with triangle constraints.The relaxation’s objective value is at most the hypergraph sparsest-cut value ΦH.Source: §9 · p. 51, SDP 9.2 · p. 51
  • Rounding maps the SDP vectors to a scalar using a Gaussian vector, then selects a cut by thresholding the scalar values.The threshold is chosen so the cut’s expansion is bounded by the rounded objective.Source: SDP 9.2 · pp. 52–53
  • The rounding analysis bounds the hyperedge contribution by log r · ΦH.The proof also uses a metric embedding for vertices appearing in the demand pairs.Source: SDP 9.2 · pp. 51–52

A Hypergraph Tensor Forms

The appendix shows that connected hypergraphs can have multiple equal largest tensor eigenvalues. This rules out relating these tensor eigenvalues to expansion in the Cheeger-like manner described for graphs.

A Hypergraph Tensor Forms

  • For every k, there are connected hypergraphs with λ1 = . . . = λk.The construction joins copies of hypergraphs recursively to form a hypercube of hypergraphs.Source: A Hypergraph Tensor Forms · pp. 57–58
  • The construction begins with two copies of a complete r-uniform hypergraph and joins selected hyperedges by splitting them into halves.The recursive construction is available as long as r > 2^(2i).Source: A Hypergraph Tensor Forms · p. 58
  • The constructed connected hypergraph Hk has equal top k eigenvalues, unlike the graph condition linking equal top eigenvalues to disconnection.The paper concludes that tensor eigenvalues are not relatable to expansion in a Cheeger-like manner.Source: A Hypergraph Tensor Forms · pp. 57–58

B Examples

The examples use polynomial checks over vertex orderings to verify procedural minimizers and exhibit limits of their uniqueness and spectral interpretation. Particular hypergraphs show that later minimizers need not be uniquely determined by earlier choices or align with Laplacian eigenvectors.

Verifying Procedural Minimizers

  • The examples verify procedural minimizers by checking weighted-space orthogonality and objective values through polynomial programs over vertex orderings.The verification reduces the condition to checking P(σ) ≥ 0 for all permutations σ.Source: B Examples · p. 58

Example B.1

  • Different minimizers for γ2 can produce different values for γ3, so the sequence of procedural minimizers is not unique.The appendix reports verifying this with an example hypergraph.Source: B Examples · pp. 58–59

Example B.3

  • Evenly distributing hyperedge weight can yield a γ2 minimizer that is not an eigenvector of the resulting Laplacian operator.The appendix also contrasts even splitting of e5 with assigning its weight to {a, c}, since b is already connected to d.Source: B Examples · pp. 59–60

Example B.2

  • In one example, every minimizer attaining γ2 is a scalar multiple of f2.The equality condition is a = b = −c = −d.Source: B Examples · pp. 59–60

Example B.4

  • For every procedural-minimizer sequence in one example, f3 is neither an eigenvector of Lw nor of Πw Lw.The example verifies the first three procedural minimizers.Source: B Examples · pp. 60–61
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