Spectral Hypergraph Partitioning

 
Project contacts: Sepideh Maleki, Udit Agarwal

Overview
Hypergraphs are a generalization of graphs where each edge (hyperedge) can connect more than two nodes. A hypergraph H = (V, E) consists of a set of nodes, V, and a set of hyperedges, E, where each hyperedge is a non-empty subset of V.

In this project we will study hypergraph bipartitioning, the problem of dividing a hypergraph into almost evenly sized disjoint pieces. Hypergraph partitioning is an active area of research, central to many VLSI, linear algebra, Boolean satisfiability, and data-mining algorithms.

There are many strategies to perform hypergraph partitioning, starting from local heuristics, such as Kernighan-Lin and Fiduccia-Mattheyses, to global spectral and multi-level techniques. In this project, we will focus on the global spectral graph partitioning technique. It constructs the Laplacian matrix and uses its smallest eigenpairs to perform the partitioning.

Goals
The overall idea of spectral approach is to build a matrix, solve an eigenvalue problem, and use eigenvectors for clustering/partitioning.

Specification
These are the steps for spectral partitioning:

• Let H=(V,E) be an input hypergraph
• Let A be the adjacency matrix of G
• Let diagonal matrix D = diag(Ae)
• Set the Laplacian matrix L=D-A
• Solve the eigenvalue problem L u = λu
• Use heuristics to transform real into discrete values (Kmean clustering)

Project deliverables and deadlines:

1. (Nov 1) A clear statement in English of the cost functions you want to optimize and the algorithms for hypergraph partitioner
2. (Nov 8) A description of how you will address the eigenvalue problem
3. (Dec 6) An implementation in c++/Galois for the algorithms you choose.
4. (Dec 6) A project report, written like an ACM conference paper, that summarizes the work you did.

Papers:
Parallel Spectral Graph Partitioning