# Hypergraph partitions

We suggest a reduction of the combinatorial problem of hypergraph partitioning to a continuous optimization problem.

## Authors

• 1 publication
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• 12 publications
12/25/2020

### BiPart: A Parallel and Deterministic Multilevel Hypergraph Partitioner

Hypergraph partitioning is used in many problem domains including VLSI d...
09/05/2017

### Inhomogeneous Hypergraph Clustering with Applications

Hypergraph partitioning is an important problem in machine learning, com...
03/25/2018

### Evolutionary n-level Hypergraph Partitioning with Adaptive Coarsening

Hypergraph partitioning is an NP-hard problem that occurs in many comput...
10/26/2018

### HYPE: Massive Hypergraph Partitioning with Neighborhood Expansion

Many important real-world applications-such as social networks or distri...
07/13/2021

### The Dynamic Complexity of Acyclic Hypergraph Homomorphisms

Finding a homomorphism from some hypergraph 𝒬 (or some relational struct...
06/15/2021

### Hypergraph Dissimilarity Measures

In this paper, we propose two novel approaches for hypergraph comparison...
06/29/2021

### Distributed Matrix Tiling Using A Hypergraph Labeling Formulation

Partitioning large matrices is an important problem in distributed linea...
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This paper is based on the papers by S. Schlag et al [1], Liu et al [2], presented to us by HIT student Han Yang and discussed in October 2018 in Harbin.

Also we are aware about other papers on this topic, e.g. the survey by D. A. Papa and I. L. Markov [3].

## The work by S. Schlaget al, “k-way Hypergraph Partitioning via n-Level Recursive Bisection”

In the paper [1], a multilevel algorithm for multigraph partitioning that contracts the vertices one at a time is developed. The running time is reduced by up to two-orders of magnitude compared to a naive -level algorithm that would be adequate for ordinary graph partitioning. The overall performance is even better than the widely used hMetis hypergraph partitioner that uses a classical multilevel algorithm with few levels. Considerably larger improvements are observed for some instance classes like social networks, for bipartitioning, and for partitions with an allowed imbalance of 10%. The algorithm presented in this work forms the basis of the hypergraph partitioning framework KaHyPar (Karlsruhe Hypergraph Partitioning).

## 1. Introduction

A hypergraph is a generalization of a graph, where each (hyper)edge can connect more than two vertices. The -way partitioning problem for a hypergraph generalizes the well-known problem of graph partitioning:

How to divide the set of vertices into disjoint parts with sizes not exceeding of the average block size, while the cost function, i.e. the sum of wieghts of all hyperedges that connect different parts should be minimized.

It is known that using hyperedges makes the partition problem more difficult [4], [5].

Hypergraph partitioning (HGP) has a lot of applications. The two important areas of applications are VLSI circuit design and scientific calculations (e.g. speeding up sparse matrix-vector multiplications)

[3]. While the first one provides an example, where minor optimization can give sufficient effect, in the second one, modelling based on hypergraphs is more flexible than that based on graphs [5], [6], [7], [8].

As the hypergraph partitioning is an NP-hard problem [9] and as it is NP-hard even to find a good approximate solution for graphs [10]

, heuristic algorithms are usually used. The most often used heuristic algorithm is the multilevel paradigm, which consists of three phases: In the coarsening phase, the hypergraph is recursively coarsened to obtain a hierarchy of smaller hypergraphs that reflect the basic structure of the input. After applying an initial partitioning algorithm to the smallest hypergraph in the second phase, coarsening is undone and, at each level, a local search method is used to improve the partition induced by the coarser level.

## 2. Combinatorial formulation of the problem

Since the problem of hypergraph partitioning is formulated approximately, we suggest to replace the original problem by its approximation from the very beginning.

So, we start with a hypergraph , consisting of a finite number of vertices and a finite number of hyperedges. Each hyperedge is given by its ends, which are connected by this hyperedge, i.e. by a finite subset In particular, among the hyperedges, there may be simplest edges, that connect only two vertices, i.e. such hyperedges that .

The -partitioning problem for a hypergraph can be formulated as follows: to find subsets such that:

1. they are disjoint;

2. up to ;

3. the number of hyperedges that connect vertices from different subsets is minimal.

## 3. Reduction of the combinatorial problem to a continuous problem

Consider first the simplest case of the combinatoriaal problem, when the hypergraph is a classical one-dimensional graph, i.e. all edges are one-dimensional.

Consider then the simplex generated by vertices . Everything happens on the space . Each vertex is identified with the delta-function on with this vertex being its support. Therefore, we may replace the set of vertices by the space of functions . If the graph is partitioned into two parts, then, instead of these parts, , , we consider two functions, , such that

 f1≥0 andf2≥0;
 Suppfi=conv(Δi).

The requirement can be replaced by the requirement

 f1(x)⋅f2(x)≡0,x∈Δ,

or, approximately, by

 maxx∈Δ|f1(x)⋅f2(x)|≤ε.

The size of a part is measured by the integral

 ∫Δfi(x)dx,

which should be approximately equal to the average part size, i.e.

 ∣∣ ∣∣∫Δfi(x)dx−12#(Δ)∣∣ ∣∣≤ε.

Each edge of the graph can be described as a function on the Cartesian product . This function should approximate the edge by using the support of the function . Then the number of edges connecting the two parts can be written as

 F(x,y)=fi(x)f2(y)g(x,y).

Therefore, the problem is reduced to minimizing the integral

 Mindef⎛⎜ ⎜⎝∫(x,y)∈Δ×ΔF(x,y)dxdy⎞⎟ ⎟⎠.

Summing up, the problem reduces to the following one: Find functions and , , satisfying the conditions:

• The condition of disjointness:

 maxx∈Δ|f1(x)⋅f2(x)|≤ε.
• The condition of almost equal sizes:

 ∣∣ ∣∣∫Δfi(x)dx−12#(Δ)∣∣ ∣∣≤ε.
• minimizing the integral

 Mindef⎛⎜ ⎜⎝∫(x,y)∈Δ×ΔF(x,y)dxdy⎞⎟ ⎟⎠=Mindef⎛⎜ ⎜⎝∫(x,y)∈Δ×Δfi(x)f2(y)g(x,y)dxdy⎞⎟ ⎟⎠.

The formulation of the problem can be naturally transferred to the case of hypergraphs, where edges are replaced by hyperedges, and the number of parts can be greater than two.

## 4. Solution of the analytical problem

Note that the condition of disjointness is of different nature than the two other conditions, namely, it should be checked at each point of separately, while the two other conditions are integrals. We may replace the disjointness condition by a weaker one:

 ∫Δf1(x)f2(x)dx≤ε#(Δ).

In this way we may get a few points, where both and are not small, but the number of such points cannot be too great.

Let us also replace by , and the problem reduces to that of finding a function such that satisfies the two conditions:

•  ∫Δf(x)(1−f(x))dx≤ε#(Δ);
•  ∣∣ ∣∣∫Δf(x)dx−12#(Δ)∣∣ ∣∣≤ε,

and minimizes the integral

 Mindef⎛⎜ ⎜⎝∫(x,y)∈Δ×Δf(x)(1−f(y))g(x,y)dxdy⎞⎟ ⎟⎠.

This can be written in a matrix form. Let denote the matrix of , the vector with all coordinates equal to 1. To simplify the notation, let also Then the above conditions are:

•  ⟨f,(a−f)⟩≤2εC;
•  |⟨a,f⟩−C|≤ε;
•  ⟨f,G(a−f)⟩→min.

This can be solved by using the Lagrange multipliers method. We have to minimize the functional

 f↦⟨f,G(a−f)⟩−λ(⟨f,(a−f)⟩−2εC)−μ((⟨a,f⟩−C)2−ε2).

The critical points of this functional satisfy

 ⟨df,G(a−f)⟩−⟨f,Gdf⟩−λ⟨df,(a−f)⟩+λ⟨f,df⟩−2μ(⟨a,f⟩−C)⟨a,df⟩=0

for any .

When the matrix is symmetric (which is natural for adjacency matrices), we may rewrite this as

 ⟨df,G(a−f)−Gf−λ(a−f)+λf−2μ(⟨a,f⟩−C)a⟩=0,

hence the critical points of the functional should satisfy

 (G−λ)(a−2f)=2μ(⟨a,f⟩−C)a,

together with

 ⟨f,(a−f)⟩≤2εC

and

 |⟨a,f⟩−C|≤ε.

## References

• [1] S. Schlag, V. Henne, T. Heuer, H. Meyerhenke, P. Sanders, Ch. Schulz, -way Hypergraph Partitioning via -Level Recursive Bisection 2016 Proceedings of the Eighteenth Workshop on Algorithm Engineering and Experiments (ALENEX), p. 53–67.
• [2] H. Liu, P. LePendu, R. Jin, and D. Dou A Hypergraph-based Method for Discovering Semantically Associated Itemsets 2011 11th IEEE International Conference on Data Mining, p.398–406.
• [3] D. A. Papa and I. L. Markov Hypergraph Partitioning and Clustering, 2011 11th IEEE International Conference on Data Mining, p.398–406. In T. F. Gonzalez, editor, Handbook of Approximation Algorithms and Metaheuristics. Chapman and Hall/CRC, 2007.
• [4] C. Curino, E. Jones, Y. Zhang, and S. Madden, Schism: A Workload-driven Approach to Database Replication and Partitioning, Proceedings VLDB Endow., 3(1-2): 48–57, September 2010.
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• [7] B. Hendrickson and T. G. Kolda. Graph partitioning models for parallel computing. Parallel Computing, 26(12):1519–1534, 2000.
• [8] S. Klamt, U. Haus, and F. Theis. Hypergraphs and Cellular Networks. PLoS Comput. Biol., 5(5): e1000385, 05 2009.
• [9] T. Lengauer. Combinatorial Algorithms for Integrated Circuit Layout. John Wiley & Sons, Inc., 1990.
• [10] Thang Nguyen Bui and Curt Jones. Finding Good Approximate Vertex and Edge Partitions is NP-Hard. Information Processing Letters, 42(3): 153–59, 1992.